id sid tid token lemma pos ap-1394 1 1 acta acta PROPN ap-1394 1 2 polytechnica polytechnica PROPN ap-1394 1 3 vol vol NOUN ap-1394 1 4 . . PUNCT ap-1394 2 1 51 51 NUM ap-1394 2 2 no no NOUN ap-1394 2 3 . . PUNCT ap-1394 3 1 4/2011 4/2011 NUM ap-1394 3 2 coquaternionic coquaternionic ADJ ap-1394 3 3 quantum quantum NOUN ap-1394 3 4 dynamics dynamic NOUN ap-1394 3 5 for for ADP ap-1394 3 6 two two NUM ap-1394 3 7 - - PUNCT ap-1394 3 8 level level NOUN ap-1394 3 9 systems system NOUN ap-1394 3 10 d. d. PROPN ap-1394 3 11 c. c. PROPN ap-1394 3 12 brody brody PROPN ap-1394 3 13 , , PUNCT ap-1394 3 14 e. e. PROPN ap-1394 3 15 m. m. PROPN ap-1394 3 16 graefe graefe PROPN ap-1394 3 17 abstract abstract VERB ap-1394 3 18 the the DET ap-1394 3 19 dynamical dynamical ADJ ap-1394 3 20 aspects aspect NOUN ap-1394 3 21 of of ADP ap-1394 3 22 a a DET ap-1394 3 23 spin12 spin12 NOUN ap-1394 3 24 particle particle NOUN ap-1394 3 25 in in ADP ap-1394 3 26 hermitian hermitian ADJ ap-1394 3 27 coquaternionic coquaternionic ADJ ap-1394 3 28 quantum quantum NOUN ap-1394 3 29 theory theory NOUN ap-1394 3 30 are be AUX ap-1394 3 31 investigated investigate VERB ap-1394 3 32 . . PUNCT ap-1394 4 1 it it PRON ap-1394 4 2 is be AUX ap-1394 4 3 shown show VERB ap-1394 4 4 that that SCONJ ap-1394 4 5 the the DET ap-1394 4 6 time time NOUN ap-1394 4 7 evolution evolution NOUN ap-1394 4 8 exhibits exhibit VERB ap-1394 4 9 three three NUM ap-1394 4 10 different different ADJ ap-1394 4 11 characteristics characteristic NOUN ap-1394 4 12 , , PUNCT ap-1394 4 13 depending depend VERB ap-1394 4 14 on on ADP ap-1394 4 15 the the DET ap-1394 4 16 values value NOUN ap-1394 4 17 of of ADP ap-1394 4 18 the the DET ap-1394 4 19 parameters parameter NOUN ap-1394 4 20 of of ADP ap-1394 4 21 the the DET ap-1394 4 22 hamiltonian hamiltonian NOUN ap-1394 4 23 . . PUNCT ap-1394 5 1 when when SCONJ ap-1394 5 2 energy energy NOUN ap-1394 5 3 eigenvalues eigenvalue NOUN ap-1394 5 4 are be AUX ap-1394 5 5 real real ADJ ap-1394 5 6 , , PUNCT ap-1394 5 7 the the DET ap-1394 5 8 evolution evolution NOUN ap-1394 5 9 is be AUX ap-1394 5 10 either either CCONJ ap-1394 5 11 isomorphic isomorphic ADJ ap-1394 5 12 to to ADP ap-1394 5 13 that that PRON ap-1394 5 14 of of ADP ap-1394 5 15 a a DET ap-1394 5 16 complex complex ADJ ap-1394 5 17 hermitian hermitian ADJ ap-1394 5 18 theory theory NOUN ap-1394 5 19 on on ADP ap-1394 5 20 a a DET ap-1394 5 21 spherical spherical ADJ ap-1394 5 22 state state NOUN ap-1394 5 23 space space NOUN ap-1394 5 24 , , PUNCT ap-1394 5 25 or or CCONJ ap-1394 5 26 else else ADV ap-1394 5 27 it it PRON ap-1394 5 28 remains remain VERB ap-1394 5 29 unitary unitary ADJ ap-1394 5 30 along along ADP ap-1394 5 31 an an DET ap-1394 5 32 open open ADJ ap-1394 5 33 orbit orbit NOUN ap-1394 5 34 on on ADP ap-1394 5 35 a a DET ap-1394 5 36 hyperbolic hyperbolic ADJ ap-1394 5 37 state state NOUN ap-1394 5 38 space space NOUN ap-1394 5 39 . . PUNCT ap-1394 6 1 when when SCONJ ap-1394 6 2 energy energy NOUN ap-1394 6 3 eigenvalues eigenvalue VERB ap-1394 6 4 form form VERB ap-1394 6 5 a a DET ap-1394 6 6 complex complex ADJ ap-1394 6 7 conjugate conjugate ADJ ap-1394 6 8 pair pair NOUN ap-1394 6 9 , , PUNCT ap-1394 6 10 the the DET ap-1394 6 11 orbit orbit NOUN ap-1394 6 12 of of ADP ap-1394 6 13 the the DET ap-1394 6 14 time time NOUN ap-1394 6 15 evolution evolution NOUN ap-1394 6 16 closes close VERB ap-1394 6 17 again again ADV ap-1394 6 18 even even ADV ap-1394 6 19 though though SCONJ ap-1394 6 20 the the DET ap-1394 6 21 state state NOUN ap-1394 6 22 space space NOUN ap-1394 6 23 is be AUX ap-1394 6 24 hyperbolic hyperbolic ADJ ap-1394 6 25 . . PUNCT ap-1394 7 1 keywords keyword NOUN ap-1394 7 2 : : PUNCT ap-1394 7 3 complexified complexifie VERB ap-1394 7 4 mechanics mechanic NOUN ap-1394 7 5 , , PUNCT ap-1394 7 6 pt pt X ap-1394 7 7 symmetry symmetry NOUN ap-1394 7 8 , , PUNCT ap-1394 7 9 hyperbolic hyperbolic ADJ ap-1394 7 10 geometry geometry NOUN ap-1394 7 11 . . PUNCT ap-1394 8 1 over over ADP ap-1394 8 2 the the DET ap-1394 8 3 last last ADJ ap-1394 8 4 decade decade NOUN ap-1394 8 5 or or CCONJ ap-1394 8 6 so so ADV ap-1394 8 7 there there PRON ap-1394 8 8 has have AUX ap-1394 8 9 been be AUX ap-1394 8 10 considerable considerable ADJ ap-1394 8 11 interest interest NOUN ap-1394 8 12 in in ADP ap-1394 8 13 the the DET ap-1394 8 14 study study NOUN ap-1394 8 15 of of ADP ap-1394 8 16 complexified complexifie VERB ap-1394 8 17 dynamical dynamical ADJ ap-1394 8 18 systems system NOUN ap-1394 8 19 ; ; PUNCT ap-1394 8 20 both both PRON ap-1394 8 21 classically classically ADV ap-1394 8 22 [ [ X ap-1394 8 23 1–6 1–6 X ap-1394 8 24 ] ] X ap-1394 8 25 and and CCONJ ap-1394 8 26 quantum quantum NOUN ap-1394 8 27 mechanically mechanically ADV ap-1394 8 28 [ [ X ap-1394 8 29 7–17 7–17 NOUN ap-1394 8 30 ] ] PUNCT ap-1394 8 31 . . PUNCT ap-1394 9 1 for for ADP ap-1394 9 2 a a DET ap-1394 9 3 classical classical ADJ ap-1394 9 4 system system NOUN ap-1394 9 5 , , PUNCT ap-1394 9 6 its its PRON ap-1394 9 7 complex complex ADJ ap-1394 9 8 extension extension NOUN ap-1394 9 9 typically typically ADV ap-1394 9 10 involves involve VERB ap-1394 9 11 the the DET ap-1394 9 12 use use NOUN ap-1394 9 13 of of ADP ap-1394 9 14 complex complex ADJ ap-1394 9 15 phase phase NOUN ap-1394 9 16 - - PUNCT ap-1394 9 17 space space NOUN ap-1394 9 18 variables variable NOUN ap-1394 9 19 : : PUNCT ap-1394 9 20 ( ( PUNCT ap-1394 9 21 x x X ap-1394 9 22 , , PUNCT ap-1394 9 23 p p NOUN ap-1394 9 24 ) ) PUNCT ap-1394 9 25 → → PUNCT ap-1394 9 26 ( ( PUNCT ap-1394 9 27 x0 x0 PROPN ap-1394 9 28 + + CCONJ ap-1394 9 29 ix1 ix1 NOUN ap-1394 9 30 , , PUNCT ap-1394 9 31 p0 p0 NOUN ap-1394 9 32 + + CCONJ ap-1394 9 33 ip1 ip1 NOUN ap-1394 9 34 ) ) PUNCT ap-1394 9 35 . . PUNCT ap-1394 10 1 hence hence ADV ap-1394 10 2 the the DET ap-1394 10 3 dimensionality dimensionality NOUN ap-1394 10 4 of of ADP ap-1394 10 5 the the DET ap-1394 10 6 phase phase NOUN ap-1394 10 7 space space NOUN ap-1394 10 8 , , PUNCT ap-1394 10 9 i.e. i.e. X ap-1394 10 10 the the DET ap-1394 10 11 dynamical dynamical ADJ ap-1394 10 12 degrees degree NOUN ap-1394 10 13 of of ADP ap-1394 10 14 freedom freedom NOUN ap-1394 10 15 , , PUNCT ap-1394 10 16 is be AUX ap-1394 10 17 doubled double VERB ap-1394 10 18 , , PUNCT ap-1394 10 19 and and CCONJ ap-1394 10 20 the the DET ap-1394 10 21 hamiltonian hamiltonian ADJ ap-1394 10 22 h(x h(x PROPN ap-1394 10 23 , , PUNCT ap-1394 10 24 p p NOUN ap-1394 10 25 ) ) PUNCT ap-1394 10 26 in in ADP ap-1394 10 27 general general ADJ ap-1394 10 28 also also ADV ap-1394 10 29 becomes become VERB ap-1394 10 30 complex complex ADJ ap-1394 10 31 . . PUNCT ap-1394 11 1 for for ADP ap-1394 11 2 a a DET ap-1394 11 3 quantum quantum ADJ ap-1394 11 4 system system NOUN ap-1394 11 5 , , PUNCT ap-1394 11 6 on on ADP ap-1394 11 7 the the DET ap-1394 11 8 other other ADJ ap-1394 11 9 hand hand NOUN ap-1394 11 10 , , PUNCT ap-1394 11 11 its its PRON ap-1394 11 12 complex complex ADJ ap-1394 11 13 extension extension NOUN ap-1394 11 14 typically typically ADV ap-1394 11 15 involves involve VERB ap-1394 11 16 the the DET ap-1394 11 17 use use NOUN ap-1394 11 18 of of ADP ap-1394 11 19 a a DET ap-1394 11 20 hamiltonian hamiltonian NOUN ap-1394 11 21 that that PRON ap-1394 11 22 is be AUX ap-1394 11 23 not not PART ap-1394 11 24 hermitian hermitian ADJ ap-1394 11 25 , , PUNCT ap-1394 11 26 whereas whereas SCONJ ap-1394 11 27 the the DET ap-1394 11 28 dynamical dynamical ADJ ap-1394 11 29 degrees degree NOUN ap-1394 11 30 of of ADP ap-1394 11 31 freedom freedom NOUN ap-1394 11 32 associated associate VERB ap-1394 11 33 with with ADP ap-1394 11 34 the the DET ap-1394 11 35 space space NOUN ap-1394 11 36 of of ADP ap-1394 11 37 states state NOUN ap-1394 11 38 — — PUNCT ap-1394 11 39 the the DET ap-1394 11 40 quantum quantum NOUN ap-1394 11 41 phase phase NOUN ap-1394 11 42 space space NOUN ap-1394 11 43 variables variable NOUN ap-1394 11 44 — — PUNCT ap-1394 11 45 are be AUX ap-1394 11 46 kept keep VERB ap-1394 11 47 real real ADJ ap-1394 11 48 . . PUNCT ap-1394 12 1 however however ADV ap-1394 12 2 , , PUNCT ap-1394 12 3 a a DET ap-1394 12 4 fully fully ADV ap-1394 12 5 complexified complexifie VERB ap-1394 12 6 quantum quantum NOUN ap-1394 12 7 dynamics dynamic NOUN ap-1394 12 8 , , PUNCT ap-1394 12 9 analogous analogous ADJ ap-1394 12 10 to to ADP ap-1394 12 11 its its PRON ap-1394 12 12 classical classical ADJ ap-1394 12 13 counterpart counterpart NOUN ap-1394 12 14 , , PUNCT ap-1394 12 15 can can AUX ap-1394 12 16 be be AUX ap-1394 12 17 formulated formulate VERB ap-1394 12 18 , , PUNCT ap-1394 12 19 where where SCONJ ap-1394 12 20 state state NOUN ap-1394 12 21 space space NOUN ap-1394 12 22 variables variable NOUN ap-1394 12 23 are be AUX ap-1394 12 24 also also ADV ap-1394 12 25 complexified complexifie VERB ap-1394 12 26 [ [ PUNCT ap-1394 12 27 18 18 NUM ap-1394 12 28 , , PUNCT ap-1394 12 29 19 19 NUM ap-1394 12 30 ] ] PUNCT ap-1394 12 31 . . PUNCT ap-1394 13 1 the the DET ap-1394 13 2 present present ADJ ap-1394 13 3 authors author NOUN ap-1394 13 4 recently recently ADV ap-1394 13 5 observed observe VERB ap-1394 13 6 that that SCONJ ap-1394 13 7 there there PRON ap-1394 13 8 are be VERB ap-1394 13 9 two two NUM ap-1394 13 10 natural natural ADJ ap-1394 13 11 ways way NOUN ap-1394 13 12 in in ADP ap-1394 13 13 which which PRON ap-1394 13 14 quantum quantum NOUN ap-1394 13 15 dynamics dynamic NOUN ap-1394 13 16 can can AUX ap-1394 13 17 be be AUX ap-1394 13 18 extended extend VERB ap-1394 13 19 into into ADP ap-1394 13 20 a a DET ap-1394 13 21 fully fully ADV ap-1394 13 22 complex complex ADJ ap-1394 13 23 domain domain NOUN ap-1394 13 24 [ [ X ap-1394 13 25 19 19 NUM ap-1394 13 26 ] ] PUNCT ap-1394 13 27 , , PUNCT ap-1394 13 28 where where SCONJ ap-1394 13 29 both both CCONJ ap-1394 13 30 the the DET ap-1394 13 31 hamiltonian hamiltonian NOUN ap-1394 13 32 and and CCONJ ap-1394 13 33 the the DET ap-1394 13 34 state state NOUN ap-1394 13 35 space space NOUN ap-1394 13 36 are be AUX ap-1394 13 37 complexified complexifie VERB ap-1394 13 38 . . PUNCT ap-1394 14 1 in in ADP ap-1394 14 2 short short ADJ ap-1394 14 3 , , PUNCT ap-1394 14 4 one one NUM ap-1394 14 5 way way NOUN ap-1394 14 6 is be AUX ap-1394 14 7 to to PART ap-1394 14 8 let let VERB ap-1394 14 9 the the DET ap-1394 14 10 state state NOUN ap-1394 14 11 space space NOUN ap-1394 14 12 variables variable NOUN ap-1394 14 13 and and CCONJ ap-1394 14 14 the the DET ap-1394 14 15 hamiltonian hamiltonian NOUN ap-1394 14 16 be be AUX ap-1394 14 17 quaternion quaternion NOUN ap-1394 14 18 valued value VERB ap-1394 14 19 ; ; PUNCT ap-1394 14 20 the the DET ap-1394 14 21 other other ADJ ap-1394 14 22 is be AUX ap-1394 14 23 to to PART ap-1394 14 24 let let VERB ap-1394 14 25 them they PRON ap-1394 14 26 be be AUX ap-1394 14 27 coquaternion coquaternion NOUN ap-1394 14 28 valued value VERB ap-1394 14 29 . . PUNCT ap-1394 15 1 the the DET ap-1394 15 2 former former ADJ ap-1394 15 3 is be AUX ap-1394 15 4 related relate VERB ap-1394 15 5 to to ADP ap-1394 15 6 quaternionic quaternionic ADJ ap-1394 15 7 quantum quantum ADJ ap-1394 15 8 mechanics mechanic NOUN ap-1394 15 9 of of ADP ap-1394 15 10 finkelstein finkelstein PROPN ap-1394 15 11 and and CCONJ ap-1394 15 12 others other NOUN ap-1394 15 13 [ [ X ap-1394 15 14 20 20 NUM ap-1394 15 15 , , PUNCT ap-1394 15 16 21 21 NUM ap-1394 15 17 ] ] PUNCT ap-1394 15 18 , , PUNCT ap-1394 15 19 whereas whereas SCONJ ap-1394 15 20 the the DET ap-1394 15 21 latter latter ADJ ap-1394 15 22 possesses possess VERB ap-1394 15 23 spectral spectral ADJ ap-1394 15 24 structures structure NOUN ap-1394 15 25 similar similar ADJ ap-1394 15 26 to to ADP ap-1394 15 27 those those PRON ap-1394 15 28 of of ADP ap-1394 15 29 pt pt ADJ ap-1394 15 30 - - ADJ ap-1394 15 31 symmetric symmetric ADJ ap-1394 15 32 quantum quantum NOUN ap-1394 15 33 theory theory NOUN ap-1394 15 34 of of ADP ap-1394 15 35 bender bender NOUN ap-1394 15 36 and and CCONJ ap-1394 15 37 others other NOUN ap-1394 16 1 [ [ X ap-1394 16 2 7–10 7–10 X ap-1394 16 3 ] ] X ap-1394 16 4 . . PUNCT ap-1394 17 1 the the DET ap-1394 17 2 purpose purpose NOUN ap-1394 17 3 of of ADP ap-1394 17 4 this this DET ap-1394 17 5 paper paper NOUN ap-1394 17 6 is be AUX ap-1394 17 7 to to PART ap-1394 17 8 work work VERB ap-1394 17 9 out out ADP ap-1394 17 10 in in ADP ap-1394 17 11 some some DET ap-1394 17 12 detail detail NOUN ap-1394 17 13 the the DET ap-1394 17 14 dynamics dynamic NOUN ap-1394 17 15 of of ADP ap-1394 17 16 an an DET ap-1394 17 17 elementary elementary ADJ ap-1394 17 18 quantum quantum NOUN ap-1394 17 19 system system NOUN ap-1394 17 20 of of ADP ap-1394 17 21 a a DET ap-1394 17 22 spin12 spin12 NOUN ap-1394 17 23 particle particle NOUN ap-1394 17 24 under under ADP ap-1394 17 25 a a DET ap-1394 17 26 coquaternionic coquaternionic ADJ ap-1394 17 27 extension extension NOUN ap-1394 17 28 , , PUNCT ap-1394 17 29 in in ADP ap-1394 17 30 a a DET ap-1394 17 31 manner manner NOUN ap-1394 17 32 analogous analogous ADJ ap-1394 17 33 to to ADP ap-1394 17 34 the the DET ap-1394 17 35 quaternionic quaternionic ADJ ap-1394 17 36 case case NOUN ap-1394 17 37 investigated investigate VERB ap-1394 17 38 elsewhere elsewhere ADV ap-1394 17 39 [ [ X ap-1394 17 40 22 22 NUM ap-1394 17 41 ] ] PUNCT ap-1394 17 42 . . PUNCT ap-1394 18 1 as as SCONJ ap-1394 18 2 illustrated illustrate VERB ap-1394 18 3 in in ADP ap-1394 18 4 [ [ X ap-1394 18 5 19 19 NUM ap-1394 18 6 ] ] PUNCT ap-1394 18 7 , , PUNCT ap-1394 18 8 a a DET ap-1394 18 9 coquaternionic coquaternionic ADJ ap-1394 18 10 dynamical dynamical ADJ ap-1394 18 11 system system NOUN ap-1394 18 12 arises arise VERB ap-1394 18 13 from from ADP ap-1394 18 14 the the DET ap-1394 18 15 extension extension NOUN ap-1394 18 16 of of ADP ap-1394 18 17 the the DET ap-1394 18 18 real real ADJ ap-1394 18 19 and and CCONJ ap-1394 18 20 the the DET ap-1394 18 21 imaginary imaginary ADJ ap-1394 18 22 parts part NOUN ap-1394 18 23 of of ADP ap-1394 18 24 the the DET ap-1394 18 25 state state NOUN ap-1394 18 26 vector vector NOUN ap-1394 18 27 in in ADP ap-1394 18 28 the the DET ap-1394 18 29 complex complex ADJ ap-1394 18 30 - - PUNCT ap-1394 18 31 j j NOUN ap-1394 18 32 direction direction NOUN ap-1394 18 33 , , PUNCT ap-1394 18 34 where where SCONJ ap-1394 18 35 j j PROPN ap-1394 18 36 is be AUX ap-1394 18 37 the the DET ap-1394 18 38 second second ADJ ap-1394 18 39 coquaternionic coquaternionic NOUN ap-1394 18 40 ‘ ' PUNCT ap-1394 18 41 imaginary imaginary ADJ ap-1394 18 42 ’ ' PUNCT ap-1394 18 43 unit unit NOUN ap-1394 18 44 ( ( PUNCT ap-1394 18 45 described describe VERB ap-1394 18 46 below below ADP ap-1394 18 47 ) ) PUNCT ap-1394 18 48 . . PUNCT ap-1394 19 1 the the DET ap-1394 19 2 general general ADJ ap-1394 19 3 dynamics dynamic NOUN ap-1394 19 4 is be AUX ap-1394 19 5 governed govern VERB ap-1394 19 6 by by ADP ap-1394 19 7 a a DET ap-1394 19 8 coquaternionic coquaternionic ADJ ap-1394 19 9 hermitian hermitian ADJ ap-1394 19 10 hamiltonian hamiltonian NOUN ap-1394 19 11 , , PUNCT ap-1394 19 12 whose whose DET ap-1394 19 13 eigenvalues eigenvalue NOUN ap-1394 19 14 are be AUX ap-1394 19 15 either either CCONJ ap-1394 19 16 real real ADJ ap-1394 19 17 or or CCONJ ap-1394 19 18 else else ADV ap-1394 19 19 appear appear VERB ap-1394 19 20 as as ADP ap-1394 19 21 complex complex ADJ ap-1394 19 22 conjugate conjugate ADJ ap-1394 19 23 pairs pair NOUN ap-1394 19 24 . . PUNCT ap-1394 20 1 here here ADV ap-1394 20 2 we we PRON ap-1394 20 3 examine examine VERB ap-1394 20 4 the the DET ap-1394 20 5 evolution evolution NOUN ap-1394 20 6 of of ADP ap-1394 20 7 the the DET ap-1394 20 8 expectation expectation NOUN ap-1394 20 9 values value NOUN ap-1394 20 10 of of ADP ap-1394 20 11 the the DET ap-1394 20 12 five five NUM ap-1394 20 13 pauli pauli PROPN ap-1394 20 14 matrices matrix NOUN ap-1394 20 15 generated generate VERB ap-1394 20 16 by by ADP ap-1394 20 17 a a DET ap-1394 20 18 generic generic ADJ ap-1394 20 19 2×2 2×2 NOUN ap-1394 20 20 coquaternionic coquaternionic NOUN ap-1394 20 21 hermitian hermitian ADJ ap-1394 20 22 hamiltonian hamiltonian NOUN ap-1394 20 23 . . PUNCT ap-1394 21 1 we we PRON ap-1394 21 2 shall shall AUX ap-1394 21 3 find find VERB ap-1394 21 4 that that SCONJ ap-1394 21 5 , , PUNCT ap-1394 21 6 depending depend VERB ap-1394 21 7 on on ADP ap-1394 21 8 the the DET ap-1394 21 9 values value NOUN ap-1394 21 10 of of ADP ap-1394 21 11 the the DET ap-1394 21 12 parameters parameter NOUN ap-1394 21 13 appearing appear VERB ap-1394 21 14 in in ADP ap-1394 21 15 the the DET ap-1394 21 16 hamiltonian hamiltonian NOUN ap-1394 21 17 , , PUNCT ap-1394 21 18 the the DET ap-1394 21 19 dynamics dynamic NOUN ap-1394 21 20 can can AUX ap-1394 21 21 be be AUX ap-1394 21 22 classified classify VERB ap-1394 21 23 into into ADP ap-1394 21 24 three three NUM ap-1394 21 25 cases case NOUN ap-1394 21 26 : : PUNCT ap-1394 21 27 ( ( PUNCT ap-1394 21 28 a a X ap-1394 21 29 ) ) PUNCT ap-1394 21 30 the the DET ap-1394 21 31 eigenvalues eigenvalue NOUN ap-1394 21 32 of of ADP ap-1394 21 33 h h NOUN ap-1394 21 34 are be AUX ap-1394 21 35 real real ADJ ap-1394 21 36 and and CCONJ ap-1394 21 37 the the DET ap-1394 21 38 dynamics dynamic NOUN ap-1394 21 39 is be AUX ap-1394 21 40 strongly strongly ADV ap-1394 21 41 unitary unitary ADJ ap-1394 21 42 in in ADP ap-1394 21 43 the the DET ap-1394 21 44 sense sense NOUN ap-1394 21 45 that that SCONJ ap-1394 21 46 the the DET ap-1394 21 47 ‘ ' PUNCT ap-1394 21 48 real real ADJ ap-1394 21 49 part part NOUN ap-1394 21 50 ’ ' PUNCT ap-1394 21 51 of of ADP ap-1394 21 52 the the DET ap-1394 21 53 dynamics dynamic NOUN ap-1394 21 54 on on ADP ap-1394 21 55 the the DET ap-1394 21 56 reduced reduced ADJ ap-1394 21 57 state state NOUN ap-1394 21 58 space space NOUN ap-1394 21 59 is be AUX ap-1394 21 60 indistinguishable indistinguishable ADJ ap-1394 21 61 from from ADP ap-1394 21 62 that that PRON ap-1394 21 63 generated generate VERB ap-1394 21 64 by by ADP ap-1394 21 65 a a DET ap-1394 21 66 standard standard ADJ ap-1394 21 67 complex complex ADJ ap-1394 21 68 hermitian hermitian ADJ ap-1394 21 69 hamiltonian hamiltonian NOUN ap-1394 21 70 ; ; PUNCT ap-1394 21 71 ( ( PUNCT ap-1394 21 72 b b X ap-1394 21 73 ) ) PUNCT ap-1394 21 74 the the DET ap-1394 21 75 eigenvalues eigenvalue NOUN ap-1394 21 76 of of ADP ap-1394 21 77 h h NOUN ap-1394 21 78 are be AUX ap-1394 21 79 real real ADJ ap-1394 21 80 and and CCONJ ap-1394 21 81 the the DET ap-1394 21 82 states state NOUN ap-1394 21 83 evolve evolve VERB ap-1394 21 84 unitarily unitarily ADV ap-1394 21 85 into into ADP ap-1394 21 86 infinity infinity NOUN ap-1394 21 87 without without ADP ap-1394 21 88 forming form VERB ap-1394 21 89 closed closed ADJ ap-1394 21 90 orbits orbit NOUN ap-1394 21 91 ; ; PUNCT ap-1394 21 92 and and CCONJ ap-1394 21 93 ( ( PUNCT ap-1394 21 94 c c X ap-1394 21 95 ) ) PUNCT ap-1394 21 96 the the DET ap-1394 21 97 eigenvalues eigenvalue NOUN ap-1394 21 98 of of ADP ap-1394 21 99 h h NOUN ap-1394 21 100 form form VERB ap-1394 21 101 a a DET ap-1394 21 102 complex complex ADJ ap-1394 21 103 - - PUNCT ap-1394 21 104 conjugate conjugate ADJ ap-1394 21 105 pair pair NOUN ap-1394 21 106 but but CCONJ ap-1394 21 107 the the DET ap-1394 21 108 dynamics dynamic NOUN ap-1394 21 109 remains remain VERB ap-1394 21 110 weakly weakly ADV ap-1394 21 111 unitary unitary ADJ ap-1394 21 112 in in ADP ap-1394 21 113 the the DET ap-1394 21 114 sense sense NOUN ap-1394 21 115 that that SCONJ ap-1394 21 116 the the DET ap-1394 21 117 real real ADJ ap-1394 21 118 part part NOUN ap-1394 21 119 of of ADP ap-1394 21 120 the the DET ap-1394 21 121 dynamics dynamic NOUN ap-1394 21 122 , , PUNCT ap-1394 21 123 although although SCONJ ap-1394 21 124 generating generate VERB ap-1394 21 125 closed closed ADJ ap-1394 21 126 orbits orbit NOUN ap-1394 21 127 , , PUNCT ap-1394 21 128 no no ADV ap-1394 21 129 longer long ADV ap-1394 21 130 lies lie VERB ap-1394 21 131 on on ADP ap-1394 21 132 the the DET ap-1394 21 133 state state NOUN ap-1394 21 134 space space NOUN ap-1394 21 135 of of ADP ap-1394 21 136 a a DET ap-1394 21 137 standard standard ADJ ap-1394 21 138 complex complex ADJ ap-1394 21 139 hermitian hermitian ADJ ap-1394 21 140 system system NOUN ap-1394 21 141 . . PUNCT ap-1394 22 1 interestingly interestingly ADV ap-1394 22 2 , , PUNCT ap-1394 22 3 properties property NOUN ap-1394 22 4 ( ( PUNCT ap-1394 22 5 b b NOUN ap-1394 22 6 ) ) PUNCT ap-1394 22 7 and and CCONJ ap-1394 22 8 ( ( PUNCT ap-1394 22 9 c c X ap-1394 22 10 ) ) PUNCT ap-1394 22 11 are be AUX ap-1394 22 12 in in ADP ap-1394 22 13 some some DET ap-1394 22 14 sense sense NOUN ap-1394 22 15 interchanged interchange VERB ap-1394 22 16 in in ADP ap-1394 22 17 a a DET ap-1394 22 18 typical typical ADJ ap-1394 22 19 pt pt ADJ ap-1394 22 20 - - ADJ ap-1394 22 21 symmetric symmetric ADJ ap-1394 22 22 hamiltonian hamiltonian NOUN ap-1394 22 23 where where SCONJ ap-1394 22 24 the the DET ap-1394 22 25 orbits orbit NOUN ap-1394 22 26 of of ADP ap-1394 22 27 a a DET ap-1394 22 28 spin12 spin12 ADJ ap-1394 22 29 system system NOUN ap-1394 22 30 are be AUX ap-1394 22 31 closed close VERB ap-1394 22 32 when when SCONJ ap-1394 22 33 eigenvalues eigenvalue NOUN ap-1394 22 34 are be AUX ap-1394 22 35 real real ADJ ap-1394 22 36 and and CCONJ ap-1394 22 37 open open ADJ ap-1394 22 38 otherwise otherwise ADV ap-1394 22 39 . . PUNCT ap-1394 23 1 these these DET ap-1394 23 2 characteristics characteristic NOUN ap-1394 23 3 are be AUX ap-1394 23 4 related relate VERB ap-1394 23 5 to to ADP ap-1394 23 6 the the DET ap-1394 23 7 three three NUM ap-1394 23 8 cases case NOUN ap-1394 23 9 investigated investigate VERB ap-1394 23 10 recently recently ADV ap-1394 23 11 by by ADP ap-1394 23 12 kisil kisil PROPN ap-1394 23 13 [ [ X ap-1394 23 14 23 23 NUM ap-1394 23 15 ] ] PUNCT ap-1394 23 16 in in ADP ap-1394 23 17 a a DET ap-1394 23 18 more more ADV ap-1394 23 19 general general ADJ ap-1394 23 20 context context NOUN ap-1394 23 21 of of ADP ap-1394 23 22 heisenberg heisenberg PROPN ap-1394 23 23 algebra algebra PROPN ap-1394 23 24 , , PUNCT ap-1394 23 25 based base VERB ap-1394 23 26 on on ADP ap-1394 23 27 the the DET ap-1394 23 28 use use NOUN ap-1394 23 29 of of ADP ap-1394 23 30 : : PUNCT ap-1394 23 31 ( ( PUNCT ap-1394 23 32 i i NOUN ap-1394 23 33 ) ) PUNCT ap-1394 23 34 spherical spherical ADJ ap-1394 23 35 imaginary imaginary ADJ ap-1394 23 36 unit unit NOUN ap-1394 23 37 i2 i2 PROPN ap-1394 23 38 = = PROPN ap-1394 23 39 −1 −1 PROPN ap-1394 23 40 ; ; PUNCT ap-1394 23 41 ( ( PUNCT ap-1394 23 42 ii ii NOUN ap-1394 23 43 ) ) PUNCT ap-1394 23 44 parabolic parabolic ADJ ap-1394 23 45 imaginary imaginary ADJ ap-1394 23 46 unit unit NOUN ap-1394 23 47 i2 i2 PROPN ap-1394 23 48 = = PROPN ap-1394 23 49 0 0 NUM ap-1394 23 50 ; ; PUNCT ap-1394 23 51 and and CCONJ ap-1394 23 52 ( ( PUNCT ap-1394 23 53 iii iii X ap-1394 23 54 ) ) PUNCT ap-1394 23 55 hyperbolic hyperbolic ADJ ap-1394 23 56 imaginary imaginary ADJ ap-1394 23 57 unit unit NOUN ap-1394 23 58 i2 i2 NOUN ap-1394 23 59 = = NOUN ap-1394 23 60 1 1 X ap-1394 23 61 . . PUNCT ap-1394 24 1 the the DET ap-1394 24 2 use use NOUN ap-1394 24 3 of of ADP ap-1394 24 4 coquaternionic coquaternionic ADJ ap-1394 24 5 hermitian hermitian ADJ ap-1394 24 6 hamiltonians hamiltonian NOUN ap-1394 24 7 thus thus ADV ap-1394 24 8 provides provide VERB ap-1394 24 9 a a DET ap-1394 24 10 concise concise ADJ ap-1394 24 11 way way NOUN ap-1394 24 12 of of ADP ap-1394 24 13 visualising visualise VERB ap-1394 24 14 these these DET ap-1394 24 15 different different ADJ ap-1394 24 16 aspects aspect NOUN ap-1394 24 17 of of ADP ap-1394 24 18 generalised generalised ADJ ap-1394 24 19 quantum quantum NOUN ap-1394 24 20 theory theory NOUN ap-1394 24 21 . . PUNCT ap-1394 25 1 before before SCONJ ap-1394 25 2 we we PRON ap-1394 25 3 analyse analyse VERB ap-1394 25 4 the the DET ap-1394 25 5 dynamics dynamic NOUN ap-1394 25 6 , , PUNCT ap-1394 25 7 let let VERB ap-1394 25 8 us we PRON ap-1394 25 9 begin begin VERB ap-1394 25 10 by by ADP ap-1394 25 11 briefly briefly ADV ap-1394 25 12 reviewing review VERB ap-1394 25 13 some some DET ap-1394 25 14 properties property NOUN ap-1394 25 15 of of ADP ap-1394 25 16 coquaternions coquaternion NOUN ap-1394 25 17 that that PRON ap-1394 25 18 are be AUX ap-1394 25 19 relevant relevant ADJ ap-1394 25 20 to to ADP ap-1394 25 21 the the DET ap-1394 25 22 ensuing ensue VERB ap-1394 25 23 discussion discussion NOUN ap-1394 25 24 . . PUNCT ap-1394 26 1 coquaternions coquaternion NOUN ap-1394 26 2 [ [ X ap-1394 26 3 24 24 NUM ap-1394 26 4 ] ] PUNCT ap-1394 26 5 , , PUNCT ap-1394 26 6 perhaps perhaps ADV ap-1394 26 7 more more ADV ap-1394 26 8 commonly commonly ADV ap-1394 26 9 known know VERB ap-1394 26 10 as as ADP ap-1394 26 11 split split ADJ ap-1394 26 12 quaternions quaternion NOUN ap-1394 26 13 , , PUNCT ap-1394 26 14 satisfy satisfy VERB ap-1394 26 15 the the DET ap-1394 26 16 algebraic algebraic PROPN ap-1394 26 17 relation relation PROPN ap-1394 26 18 i2 i2 PROPN ap-1394 26 19 = = SYM ap-1394 26 20 −1 −1 PROPN ap-1394 26 21 , , PUNCT ap-1394 26 22 j2 j2 PROPN ap-1394 26 23 = = SYM ap-1394 26 24 k2 k2 PROPN ap-1394 26 25 = = SYM ap-1394 26 26 ijk ijk PROPN ap-1394 26 27 = = SYM ap-1394 26 28 +1 +1 PROPN ap-1394 26 29 ( ( PUNCT ap-1394 26 30 1 1 NUM ap-1394 26 31 ) ) PUNCT ap-1394 26 32 14 14 NUM ap-1394 26 33 acta acta PROPN ap-1394 26 34 polytechnica polytechnica PROPN ap-1394 26 35 vol vol NOUN ap-1394 26 36 . . PUNCT ap-1394 27 1 51 51 NUM ap-1394 27 2 no no INTJ ap-1394 27 3 . . PUNCT ap-1394 28 1 4/2011 4/2011 NUM ap-1394 28 2 and and CCONJ ap-1394 28 3 the the DET ap-1394 28 4 skew skew ADJ ap-1394 28 5 - - PUNCT ap-1394 28 6 cyclic cyclic ADJ ap-1394 28 7 relation relation NOUN ap-1394 28 8 ij ij NOUN ap-1394 28 9 = = SYM ap-1394 28 10 −ji −ji PROPN ap-1394 28 11 = = SYM ap-1394 28 12 k k PROPN ap-1394 28 13 , , PUNCT ap-1394 28 14 jk jk X ap-1394 28 15 = = PUNCT ap-1394 28 16 −kj −kj PROPN ap-1394 29 1 = = SYM ap-1394 29 2 −i −i PROPN ap-1394 29 3 , , PUNCT ap-1394 29 4 ki ki PROPN ap-1394 29 5 = = PUNCT ap-1394 29 6 −ik −ik PROPN ap-1394 29 7 = = PUNCT ap-1394 29 8 j. j. X ap-1394 29 9 ( ( PUNCT ap-1394 29 10 2 2 NUM ap-1394 29 11 ) ) PUNCT ap-1394 29 12 the the DET ap-1394 29 13 conjugate conjugate NOUN ap-1394 29 14 of of ADP ap-1394 29 15 a a DET ap-1394 29 16 coquaternion coquaternion NOUN ap-1394 29 17 q q NOUN ap-1394 29 18 = = NOUN ap-1394 29 19 q0 q0 PROPN ap-1394 29 20 + + CCONJ ap-1394 29 21 iq1 iq1 NOUN ap-1394 29 22 + + CCONJ ap-1394 29 23 jq2 jq2 NOUN ap-1394 29 24 + + X ap-1394 29 25 kq3 kq3 PROPN ap-1394 29 26 is be AUX ap-1394 29 27 q̄ q̄ ADJ ap-1394 29 28 = = SYM ap-1394 29 29 q0 q0 PROPN ap-1394 30 1 − − PROPN ap-1394 30 2 iq1 iq1 PROPN ap-1394 30 3 − − PROPN ap-1394 31 1 jq2 jq2 PROPN ap-1394 31 2 − − PROPN ap-1394 31 3 kq3 kq3 PROPN ap-1394 31 4 . . PUNCT ap-1394 32 1 it it PRON ap-1394 32 2 follows follow VERB ap-1394 32 3 that that SCONJ ap-1394 32 4 the the DET ap-1394 32 5 squared square VERB ap-1394 32 6 modulus modulus NOUN ap-1394 32 7 of of ADP ap-1394 32 8 a a DET ap-1394 32 9 coquaternion coquaternion NOUN ap-1394 32 10 is be AUX ap-1394 32 11 indefinite indefinite ADJ ap-1394 32 12 : : PUNCT ap-1394 32 13 q̄q q̄q PROPN ap-1394 32 14 = = SYM ap-1394 32 15 q20 q20 NOUN ap-1394 32 16 + + CCONJ ap-1394 32 17 q21 q21 ADJ ap-1394 32 18 − − PROPN ap-1394 32 19 q22 q22 NOUN ap-1394 32 20 − − PROPN ap-1394 32 21 q23 q23 INTJ ap-1394 32 22 . . PUNCT ap-1394 33 1 unlike unlike ADP ap-1394 33 2 quaternions quaternion NOUN ap-1394 33 3 , , PUNCT ap-1394 33 4 a a DET ap-1394 33 5 coquaternion coquaternion NOUN ap-1394 33 6 need need AUX ap-1394 33 7 not not PART ap-1394 33 8 have have VERB ap-1394 33 9 an an DET ap-1394 33 10 inverse inverse NOUN ap-1394 33 11 q−1 q−1 NOUN ap-1394 33 12 = = PUNCT ap-1394 33 13 q̄/(q̄q q̄/(q̄q PROPN ap-1394 33 14 ) ) PUNCT ap-1394 33 15 if if SCONJ ap-1394 33 16 it it PRON ap-1394 33 17 is be AUX ap-1394 33 18 null null ADJ ap-1394 33 19 , , PUNCT ap-1394 33 20 i.e. i.e. X ap-1394 33 21 if if SCONJ ap-1394 33 22 q̄q q̄q PROPN ap-1394 33 23 = = SYM ap-1394 33 24 0 0 NUM ap-1394 33 25 . . PUNCT ap-1394 34 1 the the DET ap-1394 34 2 polar polar ADJ ap-1394 34 3 decomposition decomposition NOUN ap-1394 34 4 of of ADP ap-1394 34 5 a a DET ap-1394 34 6 coquaternion coquaternion NOUN ap-1394 34 7 is be AUX ap-1394 34 8 thus thus ADV ap-1394 34 9 more more ADV ap-1394 34 10 intricate intricate ADJ ap-1394 34 11 than than ADP ap-1394 34 12 that that PRON ap-1394 34 13 of of ADP ap-1394 34 14 a a DET ap-1394 34 15 quaternion quaternion NOUN ap-1394 34 16 . . PUNCT ap-1394 35 1 if if SCONJ ap-1394 35 2 a a DET ap-1394 35 3 coquaternion coquaternion NOUN ap-1394 35 4 q q NOUN ap-1394 35 5 has have VERB ap-1394 35 6 the the DET ap-1394 35 7 property property NOUN ap-1394 35 8 that that PRON ap-1394 35 9 q̄q q̄q PROPN ap-1394 35 10 > > X ap-1394 35 11 0 0 PUNCT ap-1394 35 12 and and CCONJ ap-1394 35 13 that that SCONJ ap-1394 35 14 its its PRON ap-1394 35 15 imaginary imaginary ADJ ap-1394 35 16 part part NOUN ap-1394 35 17 also also ADV ap-1394 35 18 has have VERB ap-1394 35 19 a a DET ap-1394 35 20 positive positive ADJ ap-1394 35 21 norm norm NOUN ap-1394 35 22 so so SCONJ ap-1394 35 23 that that DET ap-1394 35 24 q21 q21 NOUN ap-1394 35 25 − − PROPN ap-1394 35 26 q22 q22 NOUN ap-1394 35 27 − − PROPN ap-1394 35 28 q23 q23 PROPN ap-1394 35 29 > > X ap-1394 35 30 0 0 NUM ap-1394 35 31 , , PUNCT ap-1394 35 32 then then ADV ap-1394 35 33 q q X ap-1394 35 34 can can AUX ap-1394 35 35 be be AUX ap-1394 35 36 written write VERB ap-1394 35 37 in in ADP ap-1394 35 38 the the DET ap-1394 35 39 form form NOUN ap-1394 35 40 q q NOUN ap-1394 36 1 = = PUNCT ap-1394 36 2 |q|eiqθq |q|eiqθq NOUN ap-1394 37 1 = = PRON ap-1394 37 2 |q|(cos |q|(cos X ap-1394 38 1 θq θq ADV ap-1394 39 1 + + CCONJ ap-1394 39 2 iq iq VERB ap-1394 39 3 sin sin NOUN ap-1394 39 4 θq θq PROPN ap-1394 39 5 ) ) PUNCT ap-1394 39 6 , , PUNCT ap-1394 39 7 ( ( PUNCT ap-1394 39 8 3 3 X ap-1394 39 9 ) ) PUNCT ap-1394 39 10 where where SCONJ ap-1394 39 11 iq iq NOUN ap-1394 39 12 = = PUNCT ap-1394 39 13 iq1 iq1 PROPN ap-1394 40 1 + + CCONJ ap-1394 40 2 jq2 jq2 PRON ap-1394 40 3 + + CCONJ ap-1394 40 4 kq3√ kq3√ X ap-1394 40 5 q21 q21 PROPN ap-1394 40 6 − − PROPN ap-1394 40 7 q22 q22 NOUN ap-1394 40 8 − − PROPN ap-1394 40 9 q23 q23 PROPN ap-1394 40 10 and and CCONJ ap-1394 40 11 θq θq ADV ap-1394 40 12 = = PUNCT ap-1394 40 13 tan−1 tan−1 PROPN ap-1394 40 14 ( ( PUNCT ap-1394 40 15 √ √ NUM ap-1394 40 16 q21 q21 PROPN ap-1394 40 17 − − PROPN ap-1394 41 1 q22 q22 NOUN ap-1394 42 1 − − PROPN ap-1394 43 1 q23 q23 NOUN ap-1394 43 2 q0 q0 PROPN ap-1394 43 3 ) ) PUNCT ap-1394 43 4 . . PUNCT ap-1394 44 1 ( ( PUNCT ap-1394 44 2 4 4 X ap-1394 44 3 ) ) PUNCT ap-1394 44 4 that that SCONJ ap-1394 44 5 a a DET ap-1394 44 6 coquaternion coquaternion NOUN ap-1394 44 7 with with ADP ap-1394 44 8 a a DET ap-1394 44 9 ‘ ' PUNCT ap-1394 44 10 time time NOUN ap-1394 44 11 - - PUNCT ap-1394 44 12 like like ADJ ap-1394 44 13 ’ ' PUNCT ap-1394 44 14 imaginary imaginary ADJ ap-1394 44 15 part part NOUN ap-1394 44 16 admits admit VERB ap-1394 44 17 the the DET ap-1394 44 18 representation representation NOUN ap-1394 44 19 ( ( PUNCT ap-1394 44 20 3 3 X ap-1394 44 21 ) ) PUNCT ap-1394 44 22 leads lead VERB ap-1394 44 23 to to ADP ap-1394 44 24 the the DET ap-1394 44 25 strong strong ADJ ap-1394 44 26 unitary unitary ADJ ap-1394 44 27 dynamics dynamic NOUN ap-1394 44 28 generated generate VERB ap-1394 44 29 by by ADP ap-1394 44 30 a a DET ap-1394 44 31 coquaternionic coquaternionic ADJ ap-1394 44 32 hermitian hermitian ADJ ap-1394 44 33 hamiltonian hamiltonian NOUN ap-1394 44 34 . . PUNCT ap-1394 45 1 on on ADP ap-1394 45 2 the the DET ap-1394 45 3 other other ADJ ap-1394 45 4 hand hand NOUN ap-1394 45 5 , , PUNCT ap-1394 45 6 if if SCONJ ap-1394 45 7 q̄q q̄q PROPN ap-1394 45 8 > > X ap-1394 45 9 0 0 PUNCT ap-1394 46 1 but but CCONJ ap-1394 46 2 q21 q21 PROPN ap-1394 46 3 − − PROPN ap-1394 46 4 q22 q22 NOUN ap-1394 46 5 − − PROPN ap-1394 46 6 q23 q23 PROPN ap-1394 46 7 < < X ap-1394 46 8 0 0 NUM ap-1394 46 9 , , PUNCT ap-1394 46 10 i.e. i.e. X ap-1394 46 11 if if SCONJ ap-1394 46 12 the the DET ap-1394 46 13 imaginary imaginary ADJ ap-1394 46 14 part part NOUN ap-1394 46 15 of of ADP ap-1394 46 16 q q PROPN ap-1394 46 17 is be AUX ap-1394 46 18 ‘ ' PUNCT ap-1394 46 19 space space NOUN ap-1394 46 20 - - PUNCT ap-1394 46 21 like like ADJ ap-1394 46 22 ’ ' PUNCT ap-1394 46 23 , , PUNCT ap-1394 46 24 then then ADV ap-1394 46 25 q q NOUN ap-1394 46 26 = = PUNCT ap-1394 46 27 |q|eiqθq |q|eiqθq NOUN ap-1394 47 1 = = X ap-1394 47 2 |q|(cosh |q|(cosh PROPN ap-1394 47 3 θq θq ADV ap-1394 47 4 + + CCONJ ap-1394 47 5 iq iq PROPN ap-1394 47 6 sinh sinh NOUN ap-1394 47 7 θq θq PROPN ap-1394 47 8 ) ) PUNCT ap-1394 47 9 , , PUNCT ap-1394 47 10 ( ( PUNCT ap-1394 47 11 5 5 X ap-1394 47 12 ) ) PUNCT ap-1394 47 13 where where SCONJ ap-1394 47 14 iq iq NOUN ap-1394 47 15 = = PUNCT ap-1394 47 16 iq1 iq1 PROPN ap-1394 48 1 + + CCONJ ap-1394 48 2 jq2 jq2 PRON ap-1394 48 3 + + CCONJ ap-1394 48 4 kq3√ kq3√ ADV ap-1394 48 5 −q21 −q21 NOUN ap-1394 48 6 + + NUM ap-1394 48 7 q22 q22 NOUN ap-1394 49 1 + + CCONJ ap-1394 49 2 q23 q23 NOUN ap-1394 49 3 and and CCONJ ap-1394 49 4 θq θq ADP ap-1394 49 5 = = SYM ap-1394 49 6 tanh−1 tanh−1 PROPN ap-1394 49 7 ( ( PUNCT ap-1394 49 8 √ √ NUM ap-1394 49 9 −q21 −q21 NOUN ap-1394 49 10 + + NUM ap-1394 49 11 q22 q22 NOUN ap-1394 49 12 + + CCONJ ap-1394 49 13 q23 q23 PRON ap-1394 49 14 |q0| |q0| NOUN ap-1394 49 15 ) ) PUNCT ap-1394 49 16 . . PUNCT ap-1394 50 1 ( ( PUNCT ap-1394 50 2 6 6 NUM ap-1394 50 3 ) ) PUNCT ap-1394 50 4 if if SCONJ ap-1394 50 5 q̄q q̄q PROPN ap-1394 50 6 > > SYM ap-1394 50 7 0 0 PUNCT ap-1394 50 8 and and CCONJ ap-1394 50 9 q21 q21 PROPN ap-1394 50 10 − − PROPN ap-1394 50 11 q22 q22 NOUN ap-1394 50 12 − − PROPN ap-1394 50 13 q23 q23 NOUN ap-1394 50 14 = = SYM ap-1394 50 15 0 0 PROPN ap-1394 50 16 , , PUNCT ap-1394 50 17 then then ADV ap-1394 50 18 q q NOUN ap-1394 50 19 = = SYM ap-1394 50 20 q0(1 q0(1 PROPN ap-1394 50 21 + + CCONJ ap-1394 50 22 iq iq NOUN ap-1394 50 23 ) ) PUNCT ap-1394 50 24 , , PUNCT ap-1394 50 25 where where SCONJ ap-1394 50 26 iq iq PROPN ap-1394 50 27 = = PUNCT ap-1394 50 28 q−10 q−10 PROPN ap-1394 50 29 ( ( PUNCT ap-1394 50 30 iq1 iq1 PROPN ap-1394 51 1 + + CCONJ ap-1394 51 2 jq2 jq2 PROPN ap-1394 51 3 + + CCONJ ap-1394 51 4 kq3 kq3 PROPN ap-1394 51 5 ) ) PUNCT ap-1394 51 6 is be AUX ap-1394 51 7 the the DET ap-1394 51 8 null null ADJ ap-1394 51 9 pureimaginary pureimaginary PROPN ap-1394 51 10 coquaternion coquaternion NOUN ap-1394 51 11 . . PUNCT ap-1394 52 1 finally finally ADV ap-1394 52 2 , , PUNCT ap-1394 52 3 if if SCONJ ap-1394 52 4 q̄q q̄q PROPN ap-1394 52 5 < < X ap-1394 52 6 0 0 NUM ap-1394 52 7 , , PUNCT ap-1394 52 8 then then ADV ap-1394 52 9 we we PRON ap-1394 52 10 have have VERB ap-1394 52 11 q q NOUN ap-1394 52 12 = = PUNCT ap-1394 52 13 |q|eiqθq |q|eiqθq NOUN ap-1394 52 14 = = SYM ap-1394 52 15 |q|(sinh |q|(sinh ADJ ap-1394 53 1 θq θq ADV ap-1394 53 2 + + NUM ap-1394 53 3 iq iq PROPN ap-1394 53 4 cosh cosh NOUN ap-1394 53 5 θq θq PROPN ap-1394 53 6 ) ) PUNCT ap-1394 53 7 , , PUNCT ap-1394 53 8 ( ( PUNCT ap-1394 53 9 7 7 X ap-1394 53 10 ) ) PUNCT ap-1394 53 11 where where SCONJ ap-1394 53 12 iq iq NOUN ap-1394 53 13 = = PUNCT ap-1394 53 14 iq1 iq1 PROPN ap-1394 54 1 + + CCONJ ap-1394 54 2 jq2 jq2 PRON ap-1394 54 3 + + CCONJ ap-1394 54 4 kq3√−q21 kq3√−q21 PROPN ap-1394 54 5 + + CCONJ ap-1394 54 6 q22 q22 NOUN ap-1394 55 1 + + CCONJ ap-1394 55 2 q23 q23 NOUN ap-1394 55 3 and and CCONJ ap-1394 55 4 θq θq ADP ap-1394 55 5 = = SYM ap-1394 55 6 tanh−1 tanh−1 PROPN ap-1394 55 7 ( ( PUNCT ap-1394 55 8 √−q21 √−q21 NOUN ap-1394 55 9 + + NUM ap-1394 55 10 q22 q22 NOUN ap-1394 56 1 + + CCONJ ap-1394 56 2 q23 q23 NOUN ap-1394 56 3 q0 q0 PROPN ap-1394 56 4 ) ) PUNCT ap-1394 56 5 . . PUNCT ap-1394 57 1 ( ( PUNCT ap-1394 57 2 8) 8) NUM ap-1394 57 3 as as SCONJ ap-1394 57 4 indicated indicate VERB ap-1394 57 5 above above ADV ap-1394 57 6 , , PUNCT ap-1394 57 7 the the DET ap-1394 57 8 fact fact NOUN ap-1394 57 9 that that SCONJ ap-1394 57 10 the the DET ap-1394 57 11 polar polar ADJ ap-1394 57 12 decomposition decomposition NOUN ap-1394 57 13 of of ADP ap-1394 57 14 a a DET ap-1394 57 15 coquaternion coquaternion NOUN ap-1394 57 16 is be AUX ap-1394 57 17 represented represent VERB ap-1394 57 18 either either CCONJ ap-1394 57 19 in in ADP ap-1394 57 20 terms term NOUN ap-1394 57 21 of of ADP ap-1394 57 22 trigonometric trigonometric ADJ ap-1394 57 23 functions function NOUN ap-1394 57 24 or or CCONJ ap-1394 57 25 in in ADP ap-1394 57 26 terms term NOUN ap-1394 57 27 of of ADP ap-1394 57 28 hyperbolic hyperbolic ADJ ap-1394 57 29 functions function NOUN ap-1394 57 30 manifests manifest VERB ap-1394 57 31 itself itself PRON ap-1394 57 32 in in ADP ap-1394 57 33 the the DET ap-1394 57 34 intricate intricate ADJ ap-1394 57 35 mixture mixture NOUN ap-1394 57 36 of of ADP ap-1394 57 37 spherical spherical ADJ ap-1394 57 38 and and CCONJ ap-1394 57 39 hyperbolic hyperbolic ADJ ap-1394 57 40 geometries geometry NOUN ap-1394 57 41 associated associate VERB ap-1394 57 42 with with ADP ap-1394 57 43 the the DET ap-1394 57 44 state state NOUN ap-1394 57 45 space space NOUN ap-1394 57 46 of of ADP ap-1394 57 47 a a DET ap-1394 57 48 spin12 spin12 ADJ ap-1394 57 49 system system NOUN ap-1394 57 50 , , PUNCT ap-1394 57 51 as as SCONJ ap-1394 57 52 we we PRON ap-1394 57 53 shall shall AUX ap-1394 57 54 describe describe VERB ap-1394 57 55 in in ADP ap-1394 57 56 what what PRON ap-1394 57 57 follows follow VERB ap-1394 57 58 . . PUNCT ap-1394 58 1 in in ADP ap-1394 58 2 the the DET ap-1394 58 3 case case NOUN ap-1394 58 4 of of ADP ap-1394 58 5 a a DET ap-1394 58 6 coquaternionic coquaternionic ADJ ap-1394 58 7 matrix matrix NOUN ap-1394 58 8 ĥ ĥ PROPN ap-1394 58 9 , , PUNCT ap-1394 58 10 its its PRON ap-1394 58 11 hermitian hermitian ADJ ap-1394 58 12 conjugate conjugate NOUN ap-1394 58 13 ĥ† ĥ† PROPN ap-1394 58 14 is be AUX ap-1394 58 15 defined define VERB ap-1394 58 16 in in ADP ap-1394 58 17 a a DET ap-1394 58 18 manner manner NOUN ap-1394 58 19 identical identical ADJ ap-1394 58 20 to to ADP ap-1394 58 21 a a DET ap-1394 58 22 complex complex ADJ ap-1394 58 23 matrix matrix NOUN ap-1394 58 24 , , PUNCT ap-1394 58 25 i.e. i.e. X ap-1394 58 26 ĥ† ĥ† PROPN ap-1394 58 27 is be AUX ap-1394 58 28 the the DET ap-1394 58 29 coquaternionic coquaternionic ADJ ap-1394 58 30 conjugate conjugate NOUN ap-1394 58 31 of of ADP ap-1394 58 32 the the DET ap-1394 58 33 transpose transpose NOUN ap-1394 58 34 of of ADP ap-1394 58 35 ĥ ĥ PROPN ap-1394 58 36 . . PUNCT ap-1394 59 1 therefore therefore ADV ap-1394 59 2 , , PUNCT ap-1394 59 3 for for ADP ap-1394 59 4 a a DET ap-1394 59 5 coquaternionic coquaternionic ADJ ap-1394 59 6 two two NUM ap-1394 59 7 - - PUNCT ap-1394 59 8 level level NOUN ap-1394 59 9 system system NOUN ap-1394 59 10 , , PUNCT ap-1394 59 11 a a DET ap-1394 59 12 generic generic ADJ ap-1394 59 13 hermitian hermitian ADJ ap-1394 59 14 hamiltonian hamiltonian NOUN ap-1394 59 15 satisfying satisfy VERB ap-1394 59 16 ĥ† ĥ† PROPN ap-1394 59 17 = = PRON ap-1394 59 18 ĥ ĥ X ap-1394 59 19 can can AUX ap-1394 59 20 be be AUX ap-1394 59 21 expressed express VERB ap-1394 59 22 in in ADP ap-1394 59 23 the the DET ap-1394 59 24 form form NOUN ap-1394 59 25 ĥ ĥ PUNCT ap-1394 59 26 = = SYM ap-1394 59 27 u0 u0 PROPN ap-1394 59 28 � � PROPN ap-1394 59 29 + + CCONJ ap-1394 59 30 5∑ 5∑ ADJ ap-1394 59 31 l=1 l=1 NOUN ap-1394 59 32 ulσ̂l ulσ̂l NOUN ap-1394 59 33 , , PUNCT ap-1394 59 34 ( ( PUNCT ap-1394 59 35 9 9 NUM ap-1394 59 36 ) ) PUNCT ap-1394 59 37 where where SCONJ ap-1394 59 38 { { PUNCT ap-1394 59 39 ul}l=0 ul}l=0 NOUN ap-1394 59 40 .. .. SYM ap-1394 59 41 5 5 NUM ap-1394 59 42 ∈ ∈ PROPN ap-1394 59 43 � � NOUN ap-1394 59 44 , , PUNCT ap-1394 59 45 and and CCONJ ap-1394 59 46 σ̂1 σ̂1 NOUN ap-1394 59 47 = = SYM ap-1394 59 48 ( ( PUNCT ap-1394 59 49 0 0 NUM ap-1394 59 50 1 1 NUM ap-1394 59 51 1 1 NUM ap-1394 59 52 0 0 NUM ap-1394 59 53 ) ) PUNCT ap-1394 59 54 , , PUNCT ap-1394 59 55 σ̂2 σ̂2 NOUN ap-1394 59 56 = = PUNCT ap-1394 60 1 ( ( PUNCT ap-1394 60 2 0 0 NUM ap-1394 60 3 −i −i NOUN ap-1394 61 1 i i INTJ ap-1394 61 2 0 0 NUM ap-1394 61 3 ) ) PUNCT ap-1394 61 4 , , PUNCT ap-1394 61 5 σ̂3 σ̂3 PUNCT ap-1394 61 6 = = PUNCT ap-1394 62 1 ( ( PUNCT ap-1394 62 2 1 1 NUM ap-1394 62 3 0 0 NUM ap-1394 62 4 0 0 NUM ap-1394 62 5 −1 −1 NOUN ap-1394 62 6 ) ) PUNCT ap-1394 62 7 , , PUNCT ap-1394 62 8 ( ( PUNCT ap-1394 62 9 10 10 NUM ap-1394 62 10 ) ) PUNCT ap-1394 62 11 σ̂4 σ̂4 NOUN ap-1394 63 1 = = SYM ap-1394 63 2 ( ( PUNCT ap-1394 63 3 0 0 NUM ap-1394 63 4 −j −j VERB ap-1394 63 5 j j NOUN ap-1394 63 6 0 0 NUM ap-1394 63 7 ) ) PUNCT ap-1394 63 8 , , PUNCT ap-1394 63 9 σ̂5 σ̂5 PUNCT ap-1394 63 10 = = PUNCT ap-1394 63 11 ( ( PUNCT ap-1394 63 12 0 0 NUM ap-1394 63 13 −k −k PROPN ap-1394 63 14 k k PROPN ap-1394 63 15 0 0 NUM ap-1394 63 16 ) ) PUNCT ap-1394 63 17 are be AUX ap-1394 63 18 the the DET ap-1394 63 19 coquaternionic coquaternionic ADJ ap-1394 63 20 pauli pauli NOUN ap-1394 63 21 matrices matrix NOUN ap-1394 63 22 . . PUNCT ap-1394 64 1 the the DET ap-1394 64 2 eigenvalues eigenvalue NOUN ap-1394 64 3 of of ADP ap-1394 64 4 the the DET ap-1394 64 5 hamiltonian hamiltonian NOUN ap-1394 64 6 ( ( PUNCT ap-1394 64 7 9 9 NUM ap-1394 64 8 ) ) PUNCT ap-1394 64 9 are be AUX ap-1394 64 10 given give VERB ap-1394 64 11 by by ADP ap-1394 64 12 e± e± PROPN ap-1394 64 13 = = SYM ap-1394 64 14 u0 u0 PROPN ap-1394 64 15 ± ± PROPN ap-1394 64 16 √ √ PROPN ap-1394 64 17 u21 u21 PROPN ap-1394 64 18 + + CCONJ ap-1394 64 19 u22 u22 PROPN ap-1394 64 20 + + CCONJ ap-1394 64 21 u23 u23 NOUN ap-1394 64 22 − − PROPN ap-1394 64 23 u24 u24 NOUN ap-1394 64 24 − − NOUN ap-1394 64 25 u25 u25 NOUN ap-1394 64 26 . . PUNCT ap-1394 65 1 ( ( PUNCT ap-1394 65 2 11 11 NUM ap-1394 65 3 ) ) PUNCT ap-1394 65 4 thus thus ADV ap-1394 65 5 , , PUNCT ap-1394 65 6 they they PRON ap-1394 65 7 are be AUX ap-1394 65 8 both both ADV ap-1394 65 9 real real ADJ ap-1394 65 10 if if SCONJ ap-1394 65 11 u21+u22+u23 u21+u22+u23 PRON ap-1394 65 12 > > X ap-1394 65 13 u24+u25 u24+u25 ADJ ap-1394 65 14 ; ; PUNCT ap-1394 65 15 otherwise otherwise ADV ap-1394 65 16 they they PRON ap-1394 65 17 form form VERB ap-1394 65 18 a a DET ap-1394 65 19 complex complex ADJ ap-1394 65 20 conjugate conjugate ADJ ap-1394 65 21 pair pair NOUN ap-1394 65 22 . . PUNCT ap-1394 66 1 this this PRON ap-1394 66 2 , , PUNCT ap-1394 66 3 of of ADP ap-1394 66 4 course course NOUN ap-1394 66 5 , , PUNCT ap-1394 66 6 is be AUX ap-1394 66 7 a a DET ap-1394 66 8 characteristic characteristic ADJ ap-1394 66 9 feature feature NOUN ap-1394 66 10 of of ADP ap-1394 66 11 a a DET ap-1394 66 12 pt pt ADJ ap-1394 66 13 - - PUNCT ap-1394 66 14 symmetric symmetric ADJ ap-1394 66 15 hamiltonian hamiltonian NOUN ap-1394 66 16 . . PUNCT ap-1394 67 1 a a DET ap-1394 67 2 unitary unitary ADJ ap-1394 67 3 time time NOUN ap-1394 67 4 evolution evolution NOUN ap-1394 67 5 in in ADP ap-1394 67 6 a a DET ap-1394 67 7 coquaternionic coquaternionic ADJ ap-1394 67 8 quantum quantum NOUN ap-1394 67 9 theory theory NOUN ap-1394 67 10 is be AUX ap-1394 67 11 generated generate VERB ap-1394 67 12 by by ADP ap-1394 67 13 a a DET ap-1394 67 14 one one NUM ap-1394 67 15 - - PUNCT ap-1394 67 16 parameter parameter NOUN ap-1394 67 17 family family NOUN ap-1394 67 18 of of ADP ap-1394 67 19 unitary unitary ADJ ap-1394 67 20 operators operator NOUN ap-1394 67 21 e−ât e−ât PROPN ap-1394 67 22 , , PUNCT ap-1394 67 23 where where SCONJ ap-1394 67 24 â â PROPN ap-1394 67 25 is be AUX ap-1394 67 26 skewhermitian skewhermitian ADJ ap-1394 67 27 : : PUNCT ap-1394 67 28 ↠↠ADJ ap-1394 67 29 = = SYM ap-1394 67 30 −â. −â. PROPN ap-1394 67 31 as as ADP ap-1394 67 32 in in ADP ap-1394 67 33 the the DET ap-1394 67 34 case case NOUN ap-1394 67 35 of of ADP ap-1394 67 36 complex complex ADJ ap-1394 67 37 quantum quantum ADJ ap-1394 67 38 theory theory NOUN ap-1394 67 39 , , PUNCT ap-1394 67 40 we we PRON ap-1394 67 41 would would AUX ap-1394 67 42 like like VERB ap-1394 67 43 to to PART ap-1394 67 44 let let VERB ap-1394 67 45 the the DET ap-1394 67 46 hamiltonian hamiltonian ADJ ap-1394 67 47 ĥ ĥ AUX ap-1394 67 48 be be AUX ap-1394 67 49 the the DET ap-1394 67 50 generator generator NOUN ap-1394 67 51 of of ADP ap-1394 67 52 the the DET ap-1394 67 53 dynamics dynamic NOUN ap-1394 67 54 . . PUNCT ap-1394 68 1 for for ADP ap-1394 68 2 this this DET ap-1394 68 3 purpose purpose NOUN ap-1394 68 4 , , PUNCT ap-1394 68 5 let let VERB ap-1394 68 6 us we PRON ap-1394 68 7 write write VERB ap-1394 68 8 i i NOUN ap-1394 68 9 = = NOUN ap-1394 68 10 1 1 NUM ap-1394 68 11 ν ν NOUN ap-1394 68 12 ( ( PUNCT ap-1394 68 13 iu2 iu2 PROPN ap-1394 68 14 + + CCONJ ap-1394 68 15 ju4 ju4 PROPN ap-1394 68 16 + + CCONJ ap-1394 68 17 ku5 ku5 NOUN ap-1394 68 18 ) ) PUNCT ap-1394 68 19 , , PUNCT ap-1394 68 20 ( ( PUNCT ap-1394 68 21 12 12 NUM ap-1394 68 22 ) ) PUNCT ap-1394 68 23 where where SCONJ ap-1394 68 24 ν ν NOUN ap-1394 68 25 = = SYM ap-1394 68 26 √ √ PROPN ap-1394 68 27 u22 u22 PROPN ap-1394 68 28 − − PROPN ap-1394 68 29 u24 u24 PROPN ap-1394 68 30 − − PROPN ap-1394 68 31 u25 u25 NOUN ap-1394 68 32 if if SCONJ ap-1394 68 33 u24 u24 ADJ ap-1394 68 34 + + CCONJ ap-1394 68 35 u25 u25 VERB ap-1394 68 36 < < X ap-1394 68 37 u22 u22 PROPN ap-1394 68 38 , , PUNCT ap-1394 68 39 and and CCONJ ap-1394 68 40 ν ν X ap-1394 68 41 = = SYM ap-1394 68 42 √ √ ADP ap-1394 68 43 u24 u24 NOUN ap-1394 68 44 + + CCONJ ap-1394 68 45 u25 u25 VERB ap-1394 68 46 − − PROPN ap-1394 68 47 u22 u22 PROPN ap-1394 68 48 if if SCONJ ap-1394 68 49 u22 u22 PROPN ap-1394 68 50 < < X ap-1394 68 51 u24 u24 PROPN ap-1394 68 52 + + CCONJ ap-1394 68 53 u25 u25 ADJ ap-1394 68 54 . . PUNCT ap-1394 69 1 then then ADV ap-1394 69 2 we we PRON ap-1394 69 3 set set VERB ap-1394 69 4 â â PUNCT ap-1394 69 5 = = X ap-1394 69 6 iĥ iĥ PROPN ap-1394 69 7 and and CCONJ ap-1394 69 8 the the DET ap-1394 69 9 schrödinger schrödinger NOUN ap-1394 69 10 equation equation NOUN ap-1394 69 11 in in ADP ap-1394 69 12 units unit NOUN ap-1394 69 13 h̄ h̄ PUNCT ap-1394 69 14 = = SYM ap-1394 69 15 1 1 NUM ap-1394 69 16 is be AUX ap-1394 69 17 thus thus ADV ap-1394 69 18 given give VERB ap-1394 69 19 by by ADP ap-1394 69 20 ( ( PUNCT ap-1394 69 21 cf cf NOUN ap-1394 69 22 . . PUNCT ap-1394 70 1 [ [ X ap-1394 70 2 22 22 NUM ap-1394 70 3 ] ] SYM ap-1394 70 4 ) ) PUNCT ap-1394 70 5 |ψ̇ |ψ̇ PROPN ap-1394 70 6 〉 〉 PROPN ap-1394 70 7 = = SYM ap-1394 70 8 −iĥ −iĥ NUM ap-1394 70 9 |ψ |ψ PROPN ap-1394 70 10 〉 〉 PROPN ap-1394 70 11 . . PUNCT ap-1394 71 1 ( ( PUNCT ap-1394 71 2 13 13 NUM ap-1394 71 3 ) ) PUNCT ap-1394 71 4 it it PRON ap-1394 71 5 is be AUX ap-1394 71 6 worth worth ADJ ap-1394 71 7 remarking remark VERB ap-1394 71 8 that that SCONJ ap-1394 71 9 when when SCONJ ap-1394 71 10 u24+u25 u24+u25 ADV ap-1394 71 11 < < X ap-1394 71 12 u22 u22 PROPN ap-1394 71 13 we we PRON ap-1394 71 14 have have VERB ap-1394 71 15 i2 i2 PROPN ap-1394 71 16 = = SYM ap-1394 71 17 −1 −1 NOUN ap-1394 71 18 , , PUNCT ap-1394 71 19 whereas whereas SCONJ ap-1394 71 20 when when SCONJ ap-1394 71 21 u22 u22 PROPN ap-1394 71 22 < < X ap-1394 71 23 u24+u25 u24+u25 X ap-1394 71 24 we we PRON ap-1394 71 25 have have VERB ap-1394 71 26 i2 i2 PROPN ap-1394 71 27 = = SYM ap-1394 71 28 +1 +1 PROPN ap-1394 71 29 . . PROPN ap-1394 71 30 15 15 NUM ap-1394 71 31 acta acta PROPN ap-1394 71 32 polytechnica polytechnica PROPN ap-1394 71 33 vol vol NOUN ap-1394 71 34 . . PUNCT ap-1394 72 1 51 51 NUM ap-1394 72 2 no no INTJ ap-1394 72 3 . . PUNCT ap-1394 73 1 4/2011 4/2011 NUM ap-1394 73 2 in in ADP ap-1394 73 3 either either DET ap-1394 73 4 case case NOUN ap-1394 73 5 iĥ iĥ PROPN ap-1394 73 6 is be AUX ap-1394 73 7 a a DET ap-1394 73 8 skew skew ADJ ap-1394 73 9 - - PUNCT ap-1394 73 10 hermitian hermitian ADJ ap-1394 73 11 operator operator NOUN ap-1394 73 12 satisfying satisfying ADJ ap-1394 73 13 ( ( PUNCT ap-1394 73 14 iĥ)† iĥ)† PROPN ap-1394 73 15 = = SYM ap-1394 73 16 −iĥ −iĥ NOUN ap-1394 73 17 ; ; PUNCT ap-1394 73 18 thus thus ADV ap-1394 73 19 e−iĥt e−iĥt ADJ ap-1394 73 20 formally formally ADV ap-1394 73 21 generates generate VERB ap-1394 73 22 a a DET ap-1394 73 23 unitary unitary ADJ ap-1394 73 24 time time NOUN ap-1394 73 25 evolution evolution NOUN ap-1394 73 26 that that PRON ap-1394 73 27 preserves preserve VERB ap-1394 73 28 the the DET ap-1394 73 29 norm norm NOUN ap-1394 73 30 〈 〈 PROPN ap-1394 73 31 ψ|ψ ψ|ψ NOUN ap-1394 73 32 〉 〉 NOUN ap-1394 73 33 = = SYM ap-1394 74 1 ψ̄1ψ1 ψ̄1ψ1 PROPN ap-1394 74 2 + + PUNCT ap-1394 74 3 ψ̄2ψ2 ψ̄2ψ2 PROPN ap-1394 74 4 , , PUNCT ap-1394 74 5 where where SCONJ ap-1394 74 6 ψ̄ ψ̄ NOUN ap-1394 74 7 is be AUX ap-1394 74 8 the the DET ap-1394 74 9 coquaternionic coquaternionic ADJ ap-1394 74 10 conjugate conjugate NOUN ap-1394 74 11 of of ADP ap-1394 74 12 ψ ψ NOUN ap-1394 74 13 so so SCONJ ap-1394 74 14 that that SCONJ ap-1394 74 15 〈 〈 PROPN ap-1394 74 16 ψ| ψ| X ap-1394 74 17 represents represent VERB ap-1394 74 18 the the DET ap-1394 74 19 hermitian hermitian ADJ ap-1394 74 20 conjugate conjugate NOUN ap-1394 74 21 of of ADP ap-1394 74 22 |ψ |ψ PROPN ap-1394 74 23 〉 〉 PROPN ap-1394 74 24 . . PUNCT ap-1394 75 1 the the DET ap-1394 75 2 conservation conservation NOUN ap-1394 75 3 of of ADP ap-1394 75 4 the the DET ap-1394 75 5 norm norm NOUN ap-1394 75 6 can can AUX ap-1394 75 7 be be AUX ap-1394 75 8 checked check VERB ap-1394 75 9 directly directly ADV ap-1394 75 10 by by ADP ap-1394 75 11 use use NOUN ap-1394 75 12 of of ADP ap-1394 75 13 the the DET ap-1394 75 14 explicit explicit ADJ ap-1394 75 15 form form NOUN ap-1394 75 16 of of ADP ap-1394 75 17 the the DET ap-1394 75 18 schrödinger schrödinger NOUN ap-1394 75 19 equation equation NOUN ap-1394 75 20 in in ADP ap-1394 75 21 terms term NOUN ap-1394 75 22 of of ADP ap-1394 75 23 the the DET ap-1394 75 24 components component NOUN ap-1394 75 25 of of ADP ap-1394 75 26 the the DET ap-1394 75 27 state state NOUN ap-1394 75 28 vector vector NOUN ap-1394 75 29 :( :( PUNCT ap-1394 75 30 ψ̇1 ψ̇1 ADP ap-1394 75 31 ψ̇2 ψ̇2 PROPN ap-1394 75 32 ) ) PUNCT ap-1394 76 1 = = PUNCT ap-1394 76 2 ( ( PUNCT ap-1394 76 3 −(u0 −(u0 X ap-1394 76 4 + + CCONJ ap-1394 76 5 u3)iψ1 u3)iψ1 ADP ap-1394 76 6 − − ADP ap-1394 76 7 u1iψ2 u1iψ2 ADJ ap-1394 76 8 − − PROPN ap-1394 76 9 νψ2 νψ2 NOUN ap-1394 76 10 −(u0 −(u0 PROPN ap-1394 76 11 − − PROPN ap-1394 76 12 u3)iψ2 u3)iψ2 PROPN ap-1394 77 1 − − PROPN ap-1394 77 2 u1iψ1 u1iψ1 NOUN ap-1394 77 3 + + CCONJ ap-1394 77 4 νψ1 νψ1 PROPN ap-1394 77 5 ) ) PUNCT ap-1394 77 6 . . PUNCT ap-1394 78 1 ( ( PUNCT ap-1394 78 2 14 14 NUM ap-1394 78 3 ) ) PUNCT ap-1394 78 4 here here ADV ap-1394 78 5 we we PRON ap-1394 78 6 have have AUX ap-1394 78 7 assumed assume VERB ap-1394 78 8 u24 u24 NOUN ap-1394 79 1 + + CCONJ ap-1394 79 2 u25 u25 PRON ap-1394 79 3 < < X ap-1394 79 4 u22 u22 PROPN ap-1394 79 5 so so SCONJ ap-1394 79 6 that that SCONJ ap-1394 79 7 ν ν NOUN ap-1394 79 8 = = PRON ap-1394 79 9 √ √ PROPN ap-1394 79 10 u22 u22 PROPN ap-1394 79 11 − − PROPN ap-1394 79 12 u24 u24 PROPN ap-1394 79 13 − − PROPN ap-1394 79 14 u25 u25 ADJ ap-1394 80 1 ; ; PUNCT ap-1394 80 2 if if SCONJ ap-1394 80 3 u22 u22 PROPN ap-1394 80 4 < < X ap-1394 80 5 u24 u24 PROPN ap-1394 80 6 + + CCONJ ap-1394 80 7 u25 u25 VERB ap-1394 80 8 , , PUNCT ap-1394 80 9 we we PRON ap-1394 80 10 have have VERB ap-1394 80 11 ν ν NOUN ap-1394 80 12 = = NOUN ap-1394 80 13 √ √ ADP ap-1394 80 14 u24 u24 NOUN ap-1394 80 15 + + CCONJ ap-1394 80 16 u25 u25 VERB ap-1394 80 17 − − PROPN ap-1394 80 18 u22 u22 PROPN ap-1394 80 19 and and CCONJ ap-1394 80 20 the the DET ap-1394 80 21 sign sign NOUN ap-1394 80 22 of of ADP ap-1394 80 23 ν ν NOUN ap-1394 80 24 in in ADP ap-1394 80 25 ( ( PUNCT ap-1394 80 26 14 14 NUM ap-1394 80 27 ) ) PUNCT ap-1394 80 28 changes change NOUN ap-1394 80 29 . . PUNCT ap-1394 81 1 to to PART ap-1394 81 2 investigate investigate VERB ap-1394 81 3 properties property NOUN ap-1394 81 4 of of ADP ap-1394 81 5 the the DET ap-1394 81 6 unitary unitary ADJ ap-1394 81 7 dynamics dynamic NOUN ap-1394 81 8 generated generate VERB ap-1394 81 9 by by ADP ap-1394 81 10 the the DET ap-1394 81 11 hamiltonian hamiltonian NOUN ap-1394 81 12 ( ( PUNCT ap-1394 81 13 9 9 X ap-1394 81 14 ) ) PUNCT ap-1394 81 15 we we PRON ap-1394 81 16 shall shall AUX ap-1394 81 17 derive derive VERB ap-1394 81 18 the the DET ap-1394 81 19 evolution evolution NOUN ap-1394 81 20 equation equation NOUN ap-1394 81 21 satisfied satisfy VERB ap-1394 81 22 by by ADP ap-1394 81 23 what what PRON ap-1394 81 24 one one PRON ap-1394 81 25 might might AUX ap-1394 81 26 call call VERB ap-1394 81 27 a a DET ap-1394 81 28 ‘ ' PUNCT ap-1394 81 29 coquaternionic coquaternionic ADJ ap-1394 81 30 bloch bloch PROPN ap-1394 81 31 vector vector PROPN ap-1394 81 32 ’ ' PUNCT ap-1394 81 33 � � PROPN ap-1394 81 34 σ σ PROPN ap-1394 81 35 , , PUNCT ap-1394 81 36 whose whose DET ap-1394 81 37 components component NOUN ap-1394 81 38 are be AUX ap-1394 81 39 given give VERB ap-1394 81 40 by by ADP ap-1394 81 41 σl σl PRON ap-1394 81 42 = = PROPN ap-1394 81 43 〈 〈 PROPN ap-1394 81 44 ψ|σ̂l|ψ ψ|σ̂l|ψ PROPN ap-1394 81 45 〉 〉 PROPN ap-1394 81 46 〈 〈 PROPN ap-1394 81 47 ψ|ψ ψ|ψ NOUN ap-1394 81 48 〉 〉 PROPN ap-1394 81 49 , , PUNCT ap-1394 81 50 l l NOUN ap-1394 82 1 = = SYM ap-1394 82 2 1 1 NUM ap-1394 82 3 , , PUNCT ap-1394 82 4 . . PUNCT ap-1394 82 5 . . PUNCT ap-1394 83 1 . . PUNCT ap-1394 84 1 , , PUNCT ap-1394 84 2 5 5 X ap-1394 84 3 . . PUNCT ap-1394 84 4 ( ( PUNCT ap-1394 84 5 15 15 NUM ap-1394 84 6 ) ) PUNCT ap-1394 84 7 by by ADP ap-1394 84 8 differentiating differentiate VERB ap-1394 84 9 σl σl ADP ap-1394 84 10 in in ADP ap-1394 84 11 t t PROPN ap-1394 84 12 for for ADP ap-1394 84 13 each each DET ap-1394 84 14 l l NOUN ap-1394 84 15 and and CCONJ ap-1394 84 16 using use VERB ap-1394 84 17 the the DET ap-1394 84 18 dynamical dynamical ADJ ap-1394 84 19 equation equation NOUN ap-1394 84 20 ( ( PUNCT ap-1394 84 21 14 14 NUM ap-1394 84 22 ) ) PUNCT ap-1394 84 23 , , PUNCT ap-1394 84 24 we we PRON ap-1394 84 25 deduce deduce VERB ap-1394 84 26 , , PUNCT ap-1394 84 27 after after ADP ap-1394 84 28 rearrangements rearrangement NOUN ap-1394 84 29 of of ADP ap-1394 84 30 terms term NOUN ap-1394 84 31 , , PUNCT ap-1394 84 32 the the DET ap-1394 84 33 following follow VERB ap-1394 84 34 set set NOUN ap-1394 84 35 of of ADP ap-1394 84 36 generalised generalise VERB ap-1394 84 37 bloch bloch PROPN ap-1394 84 38 equations equation NOUN ap-1394 84 39 : : PUNCT ap-1394 84 40 1 1 NUM ap-1394 84 41 2 2 NUM ap-1394 84 42 σ̇1 σ̇1 NOUN ap-1394 84 43 = = PUNCT ap-1394 84 44 νσ3 νσ3 NOUN ap-1394 84 45 − − NOUN ap-1394 84 46 u3 u3 PROPN ap-1394 84 47 ν ν NOUN ap-1394 84 48 ( ( PUNCT ap-1394 84 49 u2σ2 u2σ2 X ap-1394 84 50 + + CCONJ ap-1394 84 51 u4σ4 u4σ4 X ap-1394 84 52 + + CCONJ ap-1394 84 53 u5σ5 u5σ5 NOUN ap-1394 84 54 ) ) PUNCT ap-1394 84 55 1 1 NUM ap-1394 84 56 2 2 NUM ap-1394 84 57 σ̇2 σ̇2 NOUN ap-1394 84 58 = = SYM ap-1394 84 59 1 1 NUM ap-1394 84 60 ν ν X ap-1394 84 61 ( ( PUNCT ap-1394 84 62 u2u3σ1 u2u3σ1 INTJ ap-1394 84 63 − − PROPN ap-1394 84 64 u1u2σ3 u1u2σ3 PROPN ap-1394 84 65 + + NUM ap-1394 84 66 u0u5σ4 u0u5σ4 NOUN ap-1394 84 67 − − PROPN ap-1394 84 68 u0u4σ5 u0u4σ5 PROPN ap-1394 84 69 ) ) PUNCT ap-1394 84 70 1 1 NUM ap-1394 84 71 2 2 NUM ap-1394 84 72 σ̇3 σ̇3 PROPN ap-1394 84 73 = = PUNCT ap-1394 84 74 −νσ1 −νσ1 PROPN ap-1394 84 75 + + NUM ap-1394 84 76 u1 u1 NOUN ap-1394 84 77 ν ν NOUN ap-1394 84 78 ( ( PUNCT ap-1394 84 79 u2σ2 u2σ2 X ap-1394 84 80 + + CCONJ ap-1394 84 81 u4σ4 u4σ4 X ap-1394 84 82 + + CCONJ ap-1394 84 83 u5σ5 u5σ5 NOUN ap-1394 84 84 ) ) PUNCT ap-1394 84 85 ( ( PUNCT ap-1394 84 86 16 16 NUM ap-1394 84 87 ) ) SYM ap-1394 84 88 1 1 NUM ap-1394 84 89 2 2 NUM ap-1394 84 90 σ̇4 σ̇4 PROPN ap-1394 84 91 = = SYM ap-1394 84 92 1 1 NUM ap-1394 84 93 ν ν NOUN ap-1394 84 94 ( ( PUNCT ap-1394 84 95 −u3u4σ1 −u3u4σ1 PROPN ap-1394 84 96 + + CCONJ ap-1394 84 97 u0u5σ2 u0u5σ2 ADJ ap-1394 85 1 + + CCONJ ap-1394 85 2 u1u4σ3 u1u4σ3 ADJ ap-1394 85 3 + + CCONJ ap-1394 85 4 u0u2σ5 u0u2σ5 NOUN ap-1394 85 5 ) ) PUNCT ap-1394 85 6 1 1 NUM ap-1394 85 7 2 2 NUM ap-1394 85 8 σ̇5 σ̇5 NUM ap-1394 85 9 = = SYM ap-1394 85 10 1 1 NUM ap-1394 85 11 ν ν NOUN ap-1394 85 12 ( ( PUNCT ap-1394 85 13 −u3u5σ1 −u3u5σ1 DET ap-1394 85 14 − − X ap-1394 85 15 u0u4σ2 u0u4σ2 PROPN ap-1394 85 16 + + CCONJ ap-1394 85 17 u1u5σ3 u1u5σ3 ADJ ap-1394 85 18 − − PROPN ap-1394 85 19 u0u2σ4 u0u2σ4 NOUN ap-1394 85 20 ) ) PUNCT ap-1394 85 21 , , PUNCT ap-1394 85 22 where where SCONJ ap-1394 85 23 we we PRON ap-1394 85 24 have have AUX ap-1394 85 25 assumed assume VERB ap-1394 85 26 u24 u24 NOUN ap-1394 85 27 + + CCONJ ap-1394 85 28 u25 u25 PRON ap-1394 85 29 < < X ap-1394 85 30 u22 u22 PROPN ap-1394 85 31 so so SCONJ ap-1394 85 32 that that SCONJ ap-1394 85 33 ν ν NOUN ap-1394 85 34 = = PRON ap-1394 85 35 √ √ PROPN ap-1394 85 36 u22 u22 PROPN ap-1394 85 37 − − PROPN ap-1394 85 38 u24 u24 PROPN ap-1394 85 39 − − NOUN ap-1394 85 40 u25 u25 ADJ ap-1394 85 41 . . PUNCT ap-1394 86 1 this this PRON ap-1394 86 2 is be AUX ap-1394 86 3 the the DET ap-1394 86 4 region region NOUN ap-1394 86 5 in in ADP ap-1394 86 6 the the DET ap-1394 86 7 parameter parameter NOUN ap-1394 86 8 space space NOUN ap-1394 86 9 where where SCONJ ap-1394 86 10 the the DET ap-1394 86 11 coquaternion coquaternion NOUN ap-1394 86 12 appearing appear VERB ap-1394 86 13 in in ADP ap-1394 86 14 the the DET ap-1394 86 15 hamiltonian hamiltonian NOUN ap-1394 86 16 has have VERB ap-1394 86 17 a a DET ap-1394 86 18 time time NOUN ap-1394 86 19 - - PUNCT ap-1394 86 20 like like ADJ ap-1394 86 21 imaginary imaginary ADJ ap-1394 86 22 part part NOUN ap-1394 86 23 . . PUNCT ap-1394 87 1 note note VERB ap-1394 87 2 that that SCONJ ap-1394 87 3 these these DET ap-1394 87 4 evolution evolution NOUN ap-1394 87 5 equations equation NOUN ap-1394 87 6 preserve preserve VERB ap-1394 87 7 the the DET ap-1394 87 8 condition condition NOUN ap-1394 87 9 : : PUNCT ap-1394 87 10 σ21 σ21 PROPN ap-1394 87 11 + + CCONJ ap-1394 87 12 σ22 σ22 PROPN ap-1394 87 13 + + CCONJ ap-1394 87 14 σ23 σ23 PROPN ap-1394 87 15 − − PROPN ap-1394 87 16 σ24 σ24 NOUN ap-1394 87 17 − − PROPN ap-1394 87 18 σ25 σ25 NOUN ap-1394 87 19 = = SYM ap-1394 87 20 1 1 NUM ap-1394 87 21 , , PUNCT ap-1394 87 22 ( ( PUNCT ap-1394 87 23 17 17 NUM ap-1394 87 24 ) ) PUNCT ap-1394 87 25 which which PRON ap-1394 87 26 can can AUX ap-1394 87 27 be be AUX ap-1394 87 28 viewed view VERB ap-1394 87 29 as as ADP ap-1394 87 30 the the DET ap-1394 87 31 defining define VERB ap-1394 87 32 equation equation NOUN ap-1394 87 33 for for ADP ap-1394 87 34 the the DET ap-1394 87 35 hyperbolic hyperbolic ADJ ap-1394 87 36 state state NOUN ap-1394 87 37 space space NOUN ap-1394 87 38 of of ADP ap-1394 87 39 a a DET ap-1394 87 40 coquaternionic coquaternionic ADJ ap-1394 87 41 two two NUM ap-1394 87 42 - - PUNCT ap-1394 87 43 level level NOUN ap-1394 87 44 system system NOUN ap-1394 87 45 . . PUNCT ap-1394 88 1 let let VERB ap-1394 88 2 us we PRON ap-1394 88 3 now now ADV ap-1394 88 4 show show VERB ap-1394 88 5 how how SCONJ ap-1394 88 6 the the DET ap-1394 88 7 dynamics dynamic NOUN ap-1394 88 8 can can AUX ap-1394 88 9 be be AUX ap-1394 88 10 reduced reduce VERB ap-1394 88 11 to to ADP ap-1394 88 12 three three NUM ap-1394 88 13 - - PUNCT ap-1394 88 14 dimensions dimension NOUN ap-1394 88 15 so so SCONJ ap-1394 88 16 as as SCONJ ap-1394 88 17 to to PART ap-1394 88 18 provide provide VERB ap-1394 88 19 a a DET ap-1394 88 20 more more ADV ap-1394 88 21 intuitive intuitive ADJ ap-1394 88 22 understanding understanding NOUN ap-1394 88 23 . . PUNCT ap-1394 89 1 for for ADP ap-1394 89 2 this this DET ap-1394 89 3 purpose purpose NOUN ap-1394 89 4 , , PUNCT ap-1394 89 5 we we PRON ap-1394 89 6 define define VERB ap-1394 89 7 the the DET ap-1394 89 8 three three NUM ap-1394 89 9 reduced reduce VERB ap-1394 89 10 spin spin NOUN ap-1394 89 11 variables variable NOUN ap-1394 89 12 σx σx PROPN ap-1394 90 1 = = PROPN ap-1394 90 2 σ1 σ1 PROPN ap-1394 90 3 , , PUNCT ap-1394 90 4 σy σy NOUN ap-1394 90 5 = = SYM ap-1394 90 6 1 1 NUM ap-1394 90 7 ν ν NOUN ap-1394 90 8 ( ( PUNCT ap-1394 90 9 u2σ2 u2σ2 ADP ap-1394 90 10 + + CCONJ ap-1394 90 11 u4σ4 u4σ4 X ap-1394 90 12 + + CCONJ ap-1394 90 13 u5σ5 u5σ5 NOUN ap-1394 90 14 ) ) PUNCT ap-1394 90 15 , , PUNCT ap-1394 90 16 σz σz PROPN ap-1394 90 17 = = SYM ap-1394 90 18 σ3 σ3 PROPN ap-1394 90 19 . . PUNCT ap-1394 91 1 ( ( PUNCT ap-1394 91 2 18 18 NUM ap-1394 91 3 ) ) PUNCT ap-1394 91 4 we we PRON ap-1394 91 5 can can AUX ap-1394 91 6 think think VERB ap-1394 91 7 of of ADP ap-1394 91 8 the the DET ap-1394 91 9 space space NOUN ap-1394 91 10 spanned span VERB ap-1394 91 11 by by ADP ap-1394 91 12 these these DET ap-1394 91 13 reduced reduce VERB ap-1394 91 14 spin spin NOUN ap-1394 91 15 variables variable NOUN ap-1394 91 16 as as ADP ap-1394 91 17 representing represent VERB ap-1394 91 18 the the DET ap-1394 91 19 ‘ ' PUNCT ap-1394 91 20 real real ADJ ap-1394 91 21 part part NOUN ap-1394 91 22 ’ ' PUNCT ap-1394 91 23 of of ADP ap-1394 91 24 the the DET ap-1394 91 25 state state NOUN ap-1394 91 26 space space NOUN ap-1394 91 27 ( ( PUNCT ap-1394 91 28 17 17 NUM ap-1394 91 29 ) ) PUNCT ap-1394 91 30 . . PUNCT ap-1394 92 1 then then ADV ap-1394 92 2 a a DET ap-1394 92 3 short short ADJ ap-1394 92 4 calculation calculation NOUN ap-1394 92 5 making make VERB ap-1394 92 6 use use NOUN ap-1394 92 7 of of ADP ap-1394 92 8 ( ( PUNCT ap-1394 92 9 16 16 NUM ap-1394 92 10 ) ) PUNCT ap-1394 92 11 shows show VERB ap-1394 92 12 that that SCONJ ap-1394 92 13 1 1 NUM ap-1394 92 14 2 2 NUM ap-1394 92 15 σ̇x σ̇x NOUN ap-1394 92 16 = = SYM ap-1394 92 17 νσz νσz PROPN ap-1394 92 18 − − PROPN ap-1394 92 19 u3σy u3σy NUM ap-1394 92 20 1 1 NUM ap-1394 92 21 2 2 NUM ap-1394 92 22 σ̇y σ̇y NOUN ap-1394 92 23 = = PRON ap-1394 92 24 u3σx u3σx PUNCT ap-1394 92 25 − − PROPN ap-1394 92 26 u1σz u1σz PUNCT ap-1394 92 27 ( ( PUNCT ap-1394 92 28 19 19 NUM ap-1394 92 29 ) ) SYM ap-1394 92 30 1 1 NUM ap-1394 92 31 2 2 NUM ap-1394 92 32 σ̇z σ̇z NOUN ap-1394 92 33 = = PUNCT ap-1394 92 34 u1σy u1σy PUNCT ap-1394 92 35 − − PROPN ap-1394 92 36 νσx νσx PROPN ap-1394 92 37 , , PUNCT ap-1394 92 38 or or CCONJ ap-1394 92 39 , , PUNCT ap-1394 92 40 more more ADV ap-1394 92 41 concisely concisely ADV ap-1394 92 42 , , PUNCT ap-1394 92 43 � � PROPN ap-1394 92 44 ̇σ ̇σ PUNCT ap-1394 92 45 = = SYM ap-1394 92 46 2 2 NUM ap-1394 92 47 � � PROPN ap-1394 92 48 b× b× NOUN ap-1394 92 49 � � PROPN ap-1394 92 50 σ σ NUM ap-1394 92 51 where where SCONJ ap-1394 92 52 � � PROPN ap-1394 92 53 b b NOUN ap-1394 92 54 = = PUNCT ap-1394 92 55 ( ( PUNCT ap-1394 92 56 u1 u1 PROPN ap-1394 92 57 , , PUNCT ap-1394 92 58 ν ν NOUN ap-1394 92 59 , , PUNCT ap-1394 92 60 u3 u3 NOUN ap-1394 92 61 ) ) PUNCT ap-1394 92 62 . . PUNCT ap-1394 93 1 hence hence ADV ap-1394 93 2 although although SCONJ ap-1394 93 3 the the DET ap-1394 93 4 state state NOUN ap-1394 93 5 space space NOUN ap-1394 93 6 of of ADP ap-1394 93 7 a a DET ap-1394 93 8 coquaternionic coquaternionic ADJ ap-1394 93 9 spin12 spin12 NOUN ap-1394 93 10 system system NOUN ap-1394 93 11 is be AUX ap-1394 93 12 a a DET ap-1394 93 13 hyperboloid hyperboloid ADJ ap-1394 93 14 ( ( PUNCT ap-1394 93 15 17 17 NUM ap-1394 93 16 ) ) PUNCT ap-1394 93 17 , , PUNCT ap-1394 93 18 remarkably remarkably ADV ap-1394 93 19 in in ADP ap-1394 93 20 the the DET ap-1394 93 21 region region NOUN ap-1394 93 22 u24 u24 NOUN ap-1394 93 23 + + CCONJ ap-1394 93 24 u25 u25 PRON ap-1394 93 25 < < X ap-1394 93 26 u22 u22 PROPN ap-1394 93 27 the the DET ap-1394 93 28 reduced reduced ADJ ap-1394 93 29 spin spin NOUN ap-1394 93 30 variables variable NOUN ap-1394 93 31 σx σx NOUN ap-1394 93 32 , , PUNCT ap-1394 93 33 σy σy PROPN ap-1394 93 34 , , PUNCT ap-1394 93 35 σz σz PRON ap-1394 93 36 defined define VERB ap-1394 93 37 by by ADP ap-1394 93 38 ( ( PUNCT ap-1394 93 39 18 18 NUM ap-1394 93 40 ) ) PUNCT ap-1394 93 41 obey obey VERB ap-1394 93 42 the the DET ap-1394 93 43 standard standard PROPN ap-1394 93 44 bloch bloch PROPN ap-1394 93 45 equations equation NOUN ap-1394 93 46 ( ( PUNCT ap-1394 93 47 19 19 NUM ap-1394 93 48 ) ) PUNCT ap-1394 93 49 . . PUNCT ap-1394 94 1 in in ADP ap-1394 94 2 particular particular ADJ ap-1394 94 3 , , PUNCT ap-1394 94 4 the the DET ap-1394 94 5 reduced reduce VERB ap-1394 94 6 motions motion NOUN ap-1394 94 7 are be AUX ap-1394 94 8 confined confine VERB ap-1394 94 9 to to ADP ap-1394 94 10 the the DET ap-1394 94 11 two two NUM ap-1394 94 12 sphere sphere NOUN ap-1394 94 13 s2 s2 NOUN ap-1394 94 14 : : PUNCT ap-1394 94 15 σ2x σ2x NOUN ap-1394 94 16 + + CCONJ ap-1394 94 17 σ2y σ2y NOUN ap-1394 94 18 + + CCONJ ap-1394 95 1 σ2z σ2z X ap-1394 95 2 = = SYM ap-1394 95 3 const const PROPN ap-1394 95 4 . . PUNCT ap-1394 95 5 , , PUNCT ap-1394 95 6 ( ( PUNCT ap-1394 95 7 20 20 NUM ap-1394 95 8 ) ) PUNCT ap-1394 95 9 where where SCONJ ap-1394 95 10 the the DET ap-1394 95 11 value value NOUN ap-1394 95 12 of of ADP ap-1394 95 13 the the DET ap-1394 95 14 right right ADJ ap-1394 95 15 side side NOUN ap-1394 95 16 of of ADP ap-1394 95 17 ( ( PUNCT ap-1394 95 18 20 20 NUM ap-1394 95 19 ) ) PUNCT ap-1394 95 20 depends depend VERB ap-1394 95 21 on on ADP ap-1394 95 22 the the DET ap-1394 95 23 initial initial ADJ ap-1394 95 24 condition condition NOUN ap-1394 95 25 ( ( PUNCT ap-1394 95 26 but but CCONJ ap-1394 95 27 is be AUX ap-1394 95 28 positive positive ADJ ap-1394 95 29 and and CCONJ ap-1394 95 30 is be AUX ap-1394 95 31 time time NOUN ap-1394 95 32 independent independent ADJ ap-1394 95 33 ) ) PUNCT ap-1394 95 34 . . PUNCT ap-1394 96 1 to to PART ap-1394 96 2 put put VERB ap-1394 96 3 the the DET ap-1394 96 4 matter matter NOUN ap-1394 96 5 differently differently ADV ap-1394 96 6 , , PUNCT ap-1394 96 7 in in ADP ap-1394 96 8 the the DET ap-1394 96 9 parameter parameter NOUN ap-1394 96 10 region region NOUN ap-1394 96 11 u24 u24 NOUN ap-1394 96 12 + + CCONJ ap-1394 96 13 u25 u25 PRON ap-1394 96 14 < < X ap-1394 96 15 u22 u22 PROPN ap-1394 96 16 , , PUNCT ap-1394 96 17 the the DET ap-1394 96 18 dynamics dynamic NOUN ap-1394 96 19 on on ADP ap-1394 96 20 the the DET ap-1394 96 21 reduced reduced ADJ ap-1394 96 22 state state NOUN ap-1394 96 23 space space NOUN ap-1394 96 24 s2 s2 NOUN ap-1394 96 25 induced induce VERB ap-1394 96 26 by by ADP ap-1394 96 27 a a DET ap-1394 96 28 coquaternionic coquaternionic ADJ ap-1394 96 29 hermitian hermitian ADJ ap-1394 96 30 hamiltonian hamiltonian NOUN ap-1394 96 31 is be AUX ap-1394 96 32 indistinguishable indistinguishable ADJ ap-1394 96 33 from from ADP ap-1394 96 34 the the DET ap-1394 96 35 conventional conventional ADJ ap-1394 96 36 unitary unitary ADJ ap-1394 96 37 dynamics dynamic NOUN ap-1394 96 38 generated generate VERB ap-1394 96 39 by by ADP ap-1394 96 40 a a DET ap-1394 96 41 complex complex ADJ ap-1394 96 42 hermitian hermitian ADJ ap-1394 96 43 hamiltonian hamiltonian NOUN ap-1394 96 44 . . PUNCT ap-1394 97 1 this this PRON ap-1394 97 2 corresponds correspond VERB ap-1394 97 3 to to ADP ap-1394 97 4 the the DET ap-1394 97 5 situation situation NOUN ap-1394 97 6 in in ADP ap-1394 97 7 a a DET ap-1394 97 8 pt pt ADJ ap-1394 97 9 - - ADJ ap-1394 97 10 symmetric symmetric ADJ ap-1394 97 11 quantum quantum NOUN ap-1394 97 12 theory theory NOUN ap-1394 97 13 whereby whereby SCONJ ap-1394 97 14 in in ADP ap-1394 97 15 some some DET ap-1394 97 16 regions region NOUN ap-1394 97 17 of of ADP ap-1394 97 18 the the DET ap-1394 97 19 parameter parameter NOUN ap-1394 97 20 space space NOUN ap-1394 97 21 the the DET ap-1394 97 22 hamiltonian hamiltonian NOUN ap-1394 97 23 is be AUX ap-1394 97 24 complex complex ADJ ap-1394 97 25 hermitian hermitian NOUN ap-1394 97 26 ( ( PUNCT ap-1394 97 27 e.g. e.g. ADV ap-1394 97 28 , , PUNCT ap-1394 97 29 a a DET ap-1394 97 30 harmonic harmonic ADJ ap-1394 97 31 oscillator oscillator NOUN ap-1394 97 32 in in ADP ap-1394 97 33 the the DET ap-1394 97 34 bender bender NOUN ap-1394 97 35 - - PUNCT ap-1394 97 36 boettcher boettcher PROPN ap-1394 97 37 hamiltonian hamiltonian ADJ ap-1394 97 38 family family NOUN ap-1394 97 39 h h NOUN ap-1394 97 40 = = PUNCT ap-1394 97 41 p2 p2 PROPN ap-1394 97 42 + + CCONJ ap-1394 97 43 x2(ix)ε x2(ix)ε PROPN ap-1394 98 1 [ [ X ap-1394 98 2 7 7 NUM ap-1394 98 3 ] ] PUNCT ap-1394 98 4 , , PUNCT ap-1394 98 5 or or CCONJ ap-1394 98 6 the the DET ap-1394 98 7 six six NUM ap-1394 98 8 - - PUNCT ap-1394 98 9 parameter parameter NOUN ap-1394 98 10 2 2 NUM ap-1394 98 11 × × NOUN ap-1394 98 12 2 2 NUM ap-1394 98 13 matrix matrix NOUN ap-1394 98 14 family family NOUN ap-1394 98 15 in in ADP ap-1394 98 16 [ [ X ap-1394 98 17 25 25 NUM ap-1394 98 18 ] ] PUNCT ap-1394 98 19 ) ) PUNCT ap-1394 98 20 . . PUNCT ap-1394 99 1 some some DET ap-1394 99 2 examples example NOUN ap-1394 99 3 of of ADP ap-1394 99 4 dynamical dynamical ADJ ap-1394 99 5 trajectories trajectory NOUN ap-1394 99 6 are be AUX ap-1394 99 7 sketched sketch VERB ap-1394 99 8 in in ADP ap-1394 99 9 figure figure NOUN ap-1394 99 10 1 1 NUM ap-1394 99 11 . . PUNCT ap-1394 100 1 the the DET ap-1394 100 2 evolution evolution NOUN ap-1394 100 3 of of ADP ap-1394 100 4 the the DET ap-1394 100 5 other other ADJ ap-1394 100 6 dynamical dynamical ADJ ap-1394 100 7 variables variable NOUN ap-1394 100 8 σ2 σ2 PROPN ap-1394 100 9 , , PUNCT ap-1394 100 10 σ4 σ4 NOUN ap-1394 100 11 , , PUNCT ap-1394 100 12 σ5 σ5 PROPN ap-1394 100 13 can can AUX ap-1394 100 14 be be AUX ap-1394 100 15 analysed analyse VERB ap-1394 100 16 as as SCONJ ap-1394 100 17 follows follow VERB ap-1394 100 18 . . PUNCT ap-1394 101 1 recall recall VERB ap-1394 101 2 that that SCONJ ap-1394 101 3 the the DET ap-1394 101 4 dynamics dynamic NOUN ap-1394 101 5 ( ( PUNCT ap-1394 101 6 19 19 NUM ap-1394 101 7 ) ) PUNCT ap-1394 101 8 preserves preserve VERB ap-1394 101 9 the the DET ap-1394 101 10 relation relation NOUN ap-1394 101 11 ( ( PUNCT ap-1394 101 12 20 20 NUM ap-1394 101 13 ) ) PUNCT ap-1394 101 14 . . PUNCT ap-1394 102 1 thus thus ADV ap-1394 102 2 , , PUNCT ap-1394 102 3 by by ADP ap-1394 102 4 subtracting subtract VERB ap-1394 102 5 ( ( PUNCT ap-1394 102 6 20 20 NUM ap-1394 102 7 ) ) PUNCT ap-1394 102 8 from from ADP ap-1394 102 9 ( ( PUNCT ap-1394 102 10 17 17 NUM ap-1394 102 11 ) ) PUNCT ap-1394 102 12 and and CCONJ ap-1394 102 13 rearranging rearrange VERB ap-1394 102 14 the the DET ap-1394 102 15 terms term NOUN ap-1394 102 16 we we PRON ap-1394 102 17 deduce deduce VERB ap-1394 102 18 that that PRON ap-1394 102 19 −(u2σ4 −(u2σ4 PUNCT ap-1394 102 20 + + CCONJ ap-1394 102 21 u4σ2 u4σ2 X ap-1394 102 22 ) ) PUNCT ap-1394 102 23 2 2 NUM ap-1394 102 24 + + CCONJ ap-1394 102 25 ( ( PUNCT ap-1394 102 26 u4σ5 u4σ5 INTJ ap-1394 102 27 − − NOUN ap-1394 102 28 u5σ4 u5σ4 NOUN ap-1394 102 29 ) ) PUNCT ap-1394 102 30 2 2 NUM ap-1394 102 31 −(u5σ2 −(u5σ2 PUNCT ap-1394 102 32 + + CCONJ ap-1394 102 33 u2σ5)2 u2σ5)2 ADJ ap-1394 102 34 = = ADJ ap-1394 102 35 const const NOUN ap-1394 102 36 . . PUNCT ap-1394 103 1 ( ( PUNCT ap-1394 103 2 21 21 NUM ap-1394 103 3 ) ) PUNCT ap-1394 103 4 this this PRON ap-1394 103 5 shows show VERB ap-1394 103 6 that that SCONJ ap-1394 103 7 the the DET ap-1394 103 8 evolution evolution NOUN ap-1394 103 9 of of ADP ap-1394 103 10 the the DET ap-1394 103 11 vector vector NOUN ap-1394 103 12 ( ( PUNCT ap-1394 103 13 σ2 σ2 PROPN ap-1394 103 14 , , PUNCT ap-1394 103 15 σ4 σ4 NOUN ap-1394 103 16 , , PUNCT ap-1394 103 17 σ5 σ5 PROPN ap-1394 103 18 ) ) PUNCT ap-1394 103 19 is be AUX ap-1394 103 20 confined confine VERB ap-1394 103 21 to to ADP ap-1394 103 22 a a DET ap-1394 103 23 hyperbolic hyperbolic ADJ ap-1394 103 24 cylinder cylinder NOUN ap-1394 103 25 . . PUNCT ap-1394 104 1 it it PRON ap-1394 104 2 turns turn VERB ap-1394 104 3 out out ADP ap-1394 104 4 that that SCONJ ap-1394 104 5 the the DET ap-1394 104 6 time time NOUN ap-1394 104 7 evolution evolution NOUN ap-1394 104 8 of of ADP ap-1394 104 9 these these DET ap-1394 104 10 ‘ ' PUNCT ap-1394 104 11 hidden hidden ADJ ap-1394 104 12 ’ ' PUNCT ap-1394 104 13 dynamical dynamical ADJ ap-1394 104 14 variables variable NOUN ap-1394 104 15 σ2 σ2 PROPN ap-1394 104 16 , , PUNCT ap-1394 104 17 σ4 σ4 NOUN ap-1394 104 18 , , PUNCT ap-1394 104 19 σ5 σ5 PROPN ap-1394 104 20 can can AUX ap-1394 104 21 also also ADV ap-1394 104 22 be be AUX ap-1394 104 23 represented represent VERB ap-1394 104 24 in in ADP ap-1394 104 25 a a DET ap-1394 104 26 form form NOUN ap-1394 104 27 similar similar ADJ ap-1394 104 28 to to ADP ap-1394 104 29 bloch bloch PROPN ap-1394 104 30 equations equation NOUN ap-1394 104 31 if if SCONJ ap-1394 104 32 we we PRON ap-1394 104 33 transform transform VERB ap-1394 104 34 the the DET ap-1394 104 35 variables variable NOUN ap-1394 104 36 according accord VERB ap-1394 104 37 to to ADP ap-1394 104 38 σy1 σy1 NOUN ap-1394 105 1 = = SYM ap-1394 105 2 u4σ5 u4σ5 PROPN ap-1394 105 3 − − PROPN ap-1394 105 4 u5σ4 u5σ4 NOUN ap-1394 105 5 , , PUNCT ap-1394 105 6 σy2 σy2 NOUN ap-1394 105 7 = = SYM ap-1394 105 8 u5σ2 u5σ2 X ap-1394 105 9 + + CCONJ ap-1394 105 10 u2σ5 u2σ5 NOUN ap-1394 105 11 , , PUNCT ap-1394 105 12 and and CCONJ ap-1394 105 13 σy3 σy3 PROPN ap-1394 105 14 = = SYM ap-1394 105 15 u2σ4 u2σ4 PROPN ap-1394 105 16 + + X ap-1394 105 17 u4σ2 u4σ2 NOUN ap-1394 105 18 . . PUNCT ap-1394 106 1 in in ADP ap-1394 106 2 terms term NOUN ap-1394 106 3 of of ADP ap-1394 106 4 these these DET ap-1394 106 5 auxiliary auxiliary ADJ ap-1394 106 6 variables variable NOUN ap-1394 106 7 we we PRON ap-1394 106 8 have have VERB ap-1394 106 9 16 16 NUM ap-1394 106 10 acta acta PROPN ap-1394 106 11 polytechnica polytechnica PROPN ap-1394 106 12 vol vol NOUN ap-1394 106 13 . . PUNCT ap-1394 107 1 51 51 NUM ap-1394 107 2 no no NOUN ap-1394 107 3 . . PUNCT ap-1394 108 1 4/2011 4/2011 NUM ap-1394 108 2 fig fig NOUN ap-1394 108 3 . . PUNCT ap-1394 109 1 1 1 NUM ap-1394 109 2 : : PUNCT ap-1394 109 3 ( ( PUNCT ap-1394 109 4 colour colour NOUN ap-1394 109 5 online online NOUN ap-1394 109 6 ) ) PUNCT ap-1394 109 7 dynamical dynamical ADJ ap-1394 109 8 trajectories trajectory NOUN ap-1394 109 9 on on ADP ap-1394 109 10 the the DET ap-1394 109 11 reduced reduced ADJ ap-1394 109 12 state state NOUN ap-1394 109 13 spaces space NOUN ap-1394 109 14 . . PUNCT ap-1394 110 1 in in ADP ap-1394 110 2 the the DET ap-1394 110 3 parameter parameter NOUN ap-1394 110 4 region region NOUN ap-1394 110 5 u22 u22 PROPN ap-1394 110 6 > > X ap-1394 110 7 u24 u24 PROPN ap-1394 110 8 + + CCONJ ap-1394 110 9 u25 u25 VERB ap-1394 110 10 the the DET ap-1394 110 11 reduced reduced ADJ ap-1394 110 12 state state NOUN ap-1394 110 13 space space NOUN ap-1394 110 14 is be AUX ap-1394 110 15 just just ADV ap-1394 110 16 a a DET ap-1394 110 17 two two NUM ap-1394 110 18 - - PUNCT ap-1394 110 19 sphere sphere NOUN ap-1394 110 20 , , PUNCT ap-1394 110 21 upon upon SCONJ ap-1394 110 22 which which PRON ap-1394 110 23 the the DET ap-1394 110 24 dynamical dynamical ADJ ap-1394 110 25 equations equation NOUN ap-1394 110 26 ( ( PUNCT ap-1394 110 27 19 19 NUM ap-1394 110 28 ) ) PUNCT ap-1394 110 29 generate generate VERB ap-1394 110 30 rabi rabi NOUN ap-1394 110 31 oscillations oscillation NOUN ap-1394 110 32 ( ( PUNCT ap-1394 110 33 left leave VERB ap-1394 110 34 figure figure NOUN ap-1394 110 35 ) ) PUNCT ap-1394 110 36 . . PUNCT ap-1394 111 1 in in ADP ap-1394 111 2 the the DET ap-1394 111 3 parameter parameter NOUN ap-1394 111 4 region region NOUN ap-1394 111 5 u22 u22 PROPN ap-1394 111 6 < < X ap-1394 111 7 u24+u25 u24+u25 VERB ap-1394 111 8 the the DET ap-1394 111 9 reduced reduced ADJ ap-1394 111 10 state state NOUN ap-1394 111 11 space space NOUN ap-1394 111 12 is be AUX ap-1394 111 13 a a DET ap-1394 111 14 two two NUM ap-1394 111 15 - - PUNCT ap-1394 111 16 dimensional dimensional ADJ ap-1394 111 17 hyperboloid hyperboloid NOUN ap-1394 111 18 , , PUNCT ap-1394 111 19 and and CCONJ ap-1394 111 20 the the DET ap-1394 111 21 dynamical dynamical ADJ ap-1394 111 22 equations equation NOUN ap-1394 111 23 ( ( PUNCT ap-1394 111 24 26 26 NUM ap-1394 111 25 ) ) PUNCT ap-1394 111 26 generate generate VERB ap-1394 111 27 open open ADJ ap-1394 111 28 trajectories trajectory NOUN ap-1394 111 29 on on ADP ap-1394 111 30 this this DET ap-1394 111 31 hyperbolic hyperbolic ADJ ap-1394 111 32 state state NOUN ap-1394 111 33 space space NOUN ap-1394 111 34 , , PUNCT ap-1394 111 35 if if SCONJ ap-1394 111 36 the the DET ap-1394 111 37 energy energy NOUN ap-1394 111 38 eigenvalues eigenvalue NOUN ap-1394 111 39 are be AUX ap-1394 111 40 real real ADJ ap-1394 111 41 ( ( PUNCT ap-1394 111 42 right right ADJ ap-1394 111 43 figure figure NOUN ap-1394 111 44 ) ) PUNCT ap-1394 111 45 . . PUNCT ap-1394 112 1 if if SCONJ ap-1394 112 2 the the DET ap-1394 112 3 eigenvalues eigenvalue NOUN ap-1394 112 4 are be AUX ap-1394 112 5 complex complex ADJ ap-1394 112 6 , , PUNCT ap-1394 112 7 the the DET ap-1394 112 8 open open ADJ ap-1394 112 9 trajectories trajectory NOUN ap-1394 112 10 are be AUX ap-1394 112 11 rotated rotate VERB ap-1394 112 12 to to PART ap-1394 112 13 form form VERB ap-1394 112 14 hyperbolic hyperbolic ADJ ap-1394 112 15 rabi rabi NOUN ap-1394 112 16 oscillations oscillation NOUN ap-1394 112 17 1 1 NUM ap-1394 112 18 2 2 NUM ap-1394 112 19 σ̇y1 σ̇y1 NOUN ap-1394 112 20 = = PROPN ap-1394 112 21 −u0 −u0 PROPN ap-1394 112 22 ν ν NOUN ap-1394 112 23 ( ( PUNCT ap-1394 112 24 u5σy2 u5σy2 NOUN ap-1394 112 25 + + CCONJ ap-1394 112 26 u4σy3 u4σy3 NOUN ap-1394 112 27 ) ) PUNCT ap-1394 112 28 1 1 NUM ap-1394 112 29 2 2 NUM ap-1394 112 30 σ̇y2 σ̇y2 NOUN ap-1394 112 31 = = SYM ap-1394 112 32 −u0 −u0 PROPN ap-1394 112 33 ν ν NOUN ap-1394 112 34 ( ( PUNCT ap-1394 112 35 u2σy3 u2σy3 PROPN ap-1394 112 36 + + CCONJ ap-1394 112 37 u5σy1 u5σy1 ADJ ap-1394 112 38 ) ) PUNCT ap-1394 112 39 ( ( PUNCT ap-1394 112 40 22 22 NUM ap-1394 112 41 ) ) PUNCT ap-1394 112 42 1 1 NUM ap-1394 112 43 2 2 NUM ap-1394 112 44 σ̇y3 σ̇y3 PROPN ap-1394 112 45 = = SYM ap-1394 112 46 −u0 −u0 PROPN ap-1394 112 47 ν ν NOUN ap-1394 112 48 ( ( PUNCT ap-1394 112 49 u4σy1 u4σy1 ADJ ap-1394 112 50 − − PROPN ap-1394 112 51 u2σy2 u2σy2 NOUN ap-1394 112 52 ) ) PUNCT ap-1394 112 53 . . PUNCT ap-1394 113 1 it it PRON ap-1394 113 2 should should AUX ap-1394 113 3 be be AUX ap-1394 113 4 evident evident ADJ ap-1394 113 5 that that SCONJ ap-1394 113 6 these these DET ap-1394 113 7 dynamics dynamic NOUN ap-1394 113 8 are be AUX ap-1394 113 9 confined confine VERB ap-1394 113 10 to to ADP ap-1394 113 11 a a DET ap-1394 113 12 hyperboloid hyperboloid NOUN ap-1394 113 13 : : PUNCT ap-1394 114 1 −σ2y1 −σ2y1 PROPN ap-1394 114 2 + + CCONJ ap-1394 114 3 σ2y2 σ2y2 PROPN ap-1394 115 1 + + CCONJ ap-1394 115 2 σ2y3 σ2y3 X ap-1394 115 3 = = SYM ap-1394 115 4 const const NOUN ap-1394 115 5 . . PUNCT ap-1394 116 1 ( ( PUNCT ap-1394 116 2 23 23 X ap-1394 116 3 ) ) PUNCT ap-1394 116 4 note note NOUN ap-1394 116 5 , , PUNCT ap-1394 116 6 however however ADV ap-1394 116 7 , , PUNCT ap-1394 116 8 that that SCONJ ap-1394 116 9 when when SCONJ ap-1394 116 10 u0 u0 ADJ ap-1394 116 11 = = NOUN ap-1394 116 12 0 0 NUM ap-1394 116 13 we we PRON ap-1394 116 14 have have AUX ap-1394 116 15 σ̇y1 σ̇y1 NOUN ap-1394 116 16 = = X ap-1394 116 17 σ̇y2 σ̇y2 NOUN ap-1394 116 18 = = SYM ap-1394 116 19 σ̇y3 σ̇y3 PROPN ap-1394 116 20 = = NOUN ap-1394 116 21 0 0 NUM ap-1394 116 22 from from ADP ap-1394 116 23 ( ( PUNCT ap-1394 116 24 22 22 NUM ap-1394 116 25 ) ) PUNCT ap-1394 116 26 , , PUNCT ap-1394 116 27 while while SCONJ ap-1394 116 28 σ2 σ2 PROPN ap-1394 116 29 , , PUNCT ap-1394 116 30 σ4 σ4 NOUN ap-1394 116 31 , , PUNCT ap-1394 116 32 σ5 σ5 NOUN ap-1394 116 33 are be AUX ap-1394 116 34 in in ADP ap-1394 116 35 general general ADJ ap-1394 116 36 evolving evolve VERB ap-1394 116 37 in in ADP ap-1394 116 38 time time NOUN ap-1394 116 39 . . PUNCT ap-1394 117 1 hence hence ADV ap-1394 117 2 in in ADP ap-1394 117 3 transforming transform VERB ap-1394 117 4 the the DET ap-1394 117 5 variables variable NOUN ap-1394 117 6 into into ADP ap-1394 117 7 σy1 σy1 NOUN ap-1394 117 8 , , PUNCT ap-1394 117 9 σy2 σy2 NOUN ap-1394 117 10 , , PUNCT ap-1394 117 11 σy3 σy3 PROPN ap-1394 117 12 , , PUNCT ap-1394 117 13 part part NOUN ap-1394 117 14 of of ADP ap-1394 117 15 the the DET ap-1394 117 16 information information NOUN ap-1394 117 17 concerning concern VERB ap-1394 117 18 the the DET ap-1394 117 19 dynamics dynamic NOUN ap-1394 117 20 is be AUX ap-1394 117 21 lost lose VERB ap-1394 117 22 . . PUNCT ap-1394 118 1 we we PRON ap-1394 118 2 see see VERB ap-1394 118 3 from from ADP ap-1394 118 4 ( ( PUNCT ap-1394 118 5 20 20 NUM ap-1394 118 6 ) ) PUNCT ap-1394 118 7 and and CCONJ ap-1394 118 8 ( ( PUNCT ap-1394 118 9 21 21 NUM ap-1394 118 10 ) ) PUNCT ap-1394 118 11 that that SCONJ ap-1394 118 12 on on ADP ap-1394 118 13 the the DET ap-1394 118 14 ‘ ' PUNCT ap-1394 118 15 imaginary imaginary ADJ ap-1394 118 16 part part NOUN ap-1394 118 17 ’ ' PUNCT ap-1394 118 18 of of ADP ap-1394 118 19 the the DET ap-1394 118 20 state state NOUN ap-1394 118 21 space space NOUN ap-1394 118 22 the the DET ap-1394 118 23 dynamics dynamic NOUN ap-1394 118 24 is be AUX ap-1394 118 25 endowed endow VERB ap-1394 118 26 with with ADP ap-1394 118 27 hyperbolic hyperbolic ADJ ap-1394 118 28 characteristics characteristic NOUN ap-1394 118 29 , , PUNCT ap-1394 118 30 which which PRON ap-1394 118 31 nevertheless nevertheless ADV ap-1394 118 32 is be AUX ap-1394 118 33 not not PART ap-1394 118 34 visible visible ADJ ap-1394 118 35 on on ADP ap-1394 118 36 the the DET ap-1394 118 37 reduced reduced ADJ ap-1394 118 38 state state NOUN ap-1394 118 39 space space NOUN ap-1394 118 40 , , PUNCT ap-1394 118 41 or or CCONJ ap-1394 118 42 the the DET ap-1394 118 43 ‘ ' PUNCT ap-1394 118 44 real real ADJ ap-1394 118 45 part part NOUN ap-1394 118 46 ’ ' PUNCT ap-1394 118 47 of of ADP ap-1394 118 48 the the DET ap-1394 118 49 state state NOUN ap-1394 118 50 space space NOUN ap-1394 118 51 s2 s2 NOUN ap-1394 118 52 spanned span VERB ap-1394 118 53 by by ADP ap-1394 118 54 σx σx PROPN ap-1394 118 55 , , PUNCT ap-1394 118 56 σy σy PROPN ap-1394 118 57 , , PUNCT ap-1394 118 58 σz σz X ap-1394 118 59 . . PUNCT ap-1394 119 1 when when SCONJ ap-1394 119 2 u22 u22 PROPN ap-1394 119 3 = = SYM ap-1394 119 4 u24 u24 PROPN ap-1394 119 5 + + CCONJ ap-1394 119 6 u25 u25 VERB ap-1394 119 7 so so SCONJ ap-1394 119 8 that that SCONJ ap-1394 119 9 the the DET ap-1394 119 10 imaginary imaginary ADJ ap-1394 119 11 part part NOUN ap-1394 119 12 of of ADP ap-1394 119 13 the the DET ap-1394 119 14 coquaternion coquaternion NOUN ap-1394 119 15 appearing appear VERB ap-1394 119 16 in in ADP ap-1394 119 17 the the DET ap-1394 119 18 hamiltonian hamiltonian NOUN ap-1394 119 19 is be AUX ap-1394 119 20 null null ADJ ap-1394 119 21 , , PUNCT ap-1394 119 22 a a DET ap-1394 119 23 calculation calculation NOUN ap-1394 119 24 shows show VERB ap-1394 119 25 that that SCONJ ap-1394 119 26 the the DET ap-1394 119 27 reduced reduce VERB ap-1394 119 28 spin spin NOUN ap-1394 119 29 variables variable NOUN ap-1394 119 30 obey obey VERB ap-1394 119 31 the the DET ap-1394 119 32 following follow VERB ap-1394 119 33 dynamical dynamical ADJ ap-1394 119 34 equations equation NOUN ap-1394 119 35 : : PUNCT ap-1394 119 36 1 1 NUM ap-1394 119 37 2 2 NUM ap-1394 119 38 σ̇x σ̇x NOUN ap-1394 119 39 = = SYM ap-1394 119 40 −u3σy −u3σy NUM ap-1394 119 41 1 1 NUM ap-1394 119 42 2 2 NUM ap-1394 119 43 σ̇y σ̇y NOUN ap-1394 119 44 = = SYM ap-1394 120 1 −u3σx −u3σx NUM ap-1394 120 2 + + NOUN ap-1394 120 3 u1σz u1σz PUNCT ap-1394 120 4 ( ( PUNCT ap-1394 120 5 24 24 NUM ap-1394 120 6 ) ) SYM ap-1394 120 7 1 1 NUM ap-1394 120 8 2 2 NUM ap-1394 120 9 σ̇z σ̇z NOUN ap-1394 120 10 = = SYM ap-1394 120 11 u1σy u1σy PROPN ap-1394 120 12 , , PUNCT ap-1394 120 13 and and CCONJ ap-1394 120 14 preserve preserve VERB ap-1394 120 15 σ2x σ2x NOUN ap-1394 120 16 − − NOUN ap-1394 120 17 σ2y σ2y NOUN ap-1394 120 18 + + CCONJ ap-1394 120 19 σ2z σ2z INTJ ap-1394 120 20 . . PUNCT ap-1394 121 1 when when SCONJ ap-1394 121 2 u22 u22 PROPN ap-1394 121 3 < < X ap-1394 121 4 u24 u24 PROPN ap-1394 121 5 + + CCONJ ap-1394 121 6 u25 u25 NOUN ap-1394 121 7 so so SCONJ ap-1394 121 8 that that SCONJ ap-1394 121 9 the the DET ap-1394 121 10 imaginary imaginary ADJ ap-1394 121 11 part part NOUN ap-1394 121 12 of of ADP ap-1394 121 13 the the DET ap-1394 121 14 coquaternion coquaternion NOUN ap-1394 121 15 in in ADP ap-1394 121 16 the the DET ap-1394 121 17 hamiltonian hamiltonian NOUN ap-1394 121 18 is be AUX ap-1394 121 19 space space NOUN ap-1394 121 20 - - PUNCT ap-1394 121 21 like like ADJ ap-1394 121 22 , , PUNCT ap-1394 121 23 the the DET ap-1394 121 24 structure structure NOUN ap-1394 121 25 of of ADP ap-1394 121 26 the the DET ap-1394 121 27 state state NOUN ap-1394 121 28 space space NOUN ap-1394 121 29 , , PUNCT ap-1394 121 30 as as ADV ap-1394 121 31 well well ADV ap-1394 121 32 as as ADP ap-1394 121 33 the the DET ap-1394 121 34 dynamics dynamic NOUN ap-1394 121 35 , , PUNCT ap-1394 121 36 change change NOUN ap-1394 121 37 , , PUNCT ap-1394 121 38 and and CCONJ ap-1394 121 39 they they PRON ap-1394 121 40 exhibit exhibit VERB ap-1394 121 41 an an DET ap-1394 121 42 interesting interesting ADJ ap-1394 121 43 and and CCONJ ap-1394 121 44 nontrivial nontrivial ADJ ap-1394 121 45 behaviour behaviour NOUN ap-1394 121 46 . . PUNCT ap-1394 122 1 the the DET ap-1394 122 2 five five NUM ap-1394 122 3 - - PUNCT ap-1394 122 4 dimensional dimensional ADJ ap-1394 122 5 spin spin NOUN ap-1394 122 6 variables variable NOUN ap-1394 122 7 in in ADP ap-1394 122 8 this this DET ap-1394 122 9 case case NOUN ap-1394 122 10 evolve evolve VERB ap-1394 122 11 according accord VERB ap-1394 122 12 to to ADP ap-1394 122 13 1 1 NUM ap-1394 122 14 2 2 NUM ap-1394 122 15 σ̇1 σ̇1 PROPN ap-1394 122 16 = = PUNCT ap-1394 122 17 −νσ3 −νσ3 PROPN ap-1394 122 18 − − NOUN ap-1394 122 19 u3 u3 PROPN ap-1394 122 20 ν ν NOUN ap-1394 122 21 ( ( PUNCT ap-1394 122 22 u2σ2 u2σ2 X ap-1394 122 23 + + CCONJ ap-1394 122 24 u4σ4 u4σ4 X ap-1394 122 25 + + CCONJ ap-1394 122 26 u5σ5 u5σ5 NOUN ap-1394 122 27 ) ) PUNCT ap-1394 122 28 1 1 NUM ap-1394 122 29 2 2 NUM ap-1394 122 30 σ̇2 σ̇2 NOUN ap-1394 122 31 = = SYM ap-1394 122 32 1 1 NUM ap-1394 122 33 ν ν X ap-1394 122 34 ( ( PUNCT ap-1394 122 35 u2u3σ1 u2u3σ1 INTJ ap-1394 122 36 − − PROPN ap-1394 122 37 u1u2σ3 u1u2σ3 PROPN ap-1394 122 38 + + NUM ap-1394 122 39 u0u5σ4 u0u5σ4 NOUN ap-1394 122 40 − − PROPN ap-1394 122 41 u0u4σ5 u0u4σ5 PROPN ap-1394 122 42 ) ) PUNCT ap-1394 122 43 1 1 NUM ap-1394 122 44 2 2 NUM ap-1394 122 45 σ̇3 σ̇3 NOUN ap-1394 122 46 = = PUNCT ap-1394 122 47 νσ1 νσ1 VERB ap-1394 122 48 + + CCONJ ap-1394 122 49 u1 u1 NOUN ap-1394 122 50 ν ν NOUN ap-1394 122 51 ( ( PUNCT ap-1394 122 52 u2σ2 u2σ2 X ap-1394 122 53 + + CCONJ ap-1394 122 54 u4σ4 u4σ4 X ap-1394 122 55 + + CCONJ ap-1394 122 56 u5σ5 u5σ5 NOUN ap-1394 122 57 ) ) PUNCT ap-1394 122 58 ( ( PUNCT ap-1394 122 59 25 25 NUM ap-1394 122 60 ) ) PUNCT ap-1394 122 61 1 1 NUM ap-1394 122 62 2 2 NUM ap-1394 122 63 σ̇4 σ̇4 PROPN ap-1394 122 64 = = SYM ap-1394 122 65 1 1 NUM ap-1394 122 66 ν ν NOUN ap-1394 122 67 ( ( PUNCT ap-1394 122 68 −u3u4σ1 −u3u4σ1 PROPN ap-1394 122 69 + + CCONJ ap-1394 122 70 u0u5σ2 u0u5σ2 ADJ ap-1394 122 71 + + CCONJ ap-1394 122 72 u1u4σ3 u1u4σ3 ADJ ap-1394 122 73 + + CCONJ ap-1394 122 74 u0u2σ5 u0u2σ5 NOUN ap-1394 122 75 ) ) PUNCT ap-1394 122 76 1 1 NUM ap-1394 122 77 2 2 NUM ap-1394 122 78 σ̇5 σ̇5 NUM ap-1394 122 79 = = SYM ap-1394 122 80 1 1 NUM ap-1394 122 81 ν ν NOUN ap-1394 122 82 ( ( PUNCT ap-1394 122 83 −u3u5σ1 −u3u5σ1 DET ap-1394 122 84 − − X ap-1394 122 85 u0u4σ2 u0u4σ2 PROPN ap-1394 123 1 + + CCONJ ap-1394 123 2 u1u5σ3 u1u5σ3 ADJ ap-1394 123 3 − − PROPN ap-1394 123 4 u0u2σ4 u0u2σ4 NOUN ap-1394 123 5 ) ) PUNCT ap-1394 123 6 , , PUNCT ap-1394 123 7 where where SCONJ ap-1394 123 8 ν ν X ap-1394 123 9 = = SYM ap-1394 123 10 √ √ NUM ap-1394 123 11 u24 u24 NOUN ap-1394 123 12 + + CCONJ ap-1394 123 13 u25 u25 VERB ap-1394 123 14 − − PROPN ap-1394 123 15 u22 u22 PROPN ap-1394 123 16 . . PUNCT ap-1394 124 1 these these DET ap-1394 124 2 evolution evolution NOUN ap-1394 124 3 equations equation NOUN ap-1394 124 4 preserve preserve VERB ap-1394 124 5 the the DET ap-1394 124 6 normalisation normalisation NOUN ap-1394 124 7 ( ( PUNCT ap-1394 124 8 17 17 NUM ap-1394 124 9 ) ) PUNCT ap-1394 124 10 . . PUNCT ap-1394 125 1 however however ADV ap-1394 125 2 , , PUNCT ap-1394 125 3 in in ADP ap-1394 125 4 the the DET ap-1394 125 5 region region NOUN ap-1394 125 6 u22 u22 PROPN ap-1394 125 7 < < X ap-1394 125 8 u24 u24 PROPN ap-1394 125 9 + + CCONJ ap-1394 125 10 u25 u25 VERB ap-1394 125 11 the the DET ap-1394 125 12 reduced reduced ADJ ap-1394 125 13 spin spin NOUN ap-1394 125 14 variables variable NOUN ap-1394 125 15 σx σx NOUN ap-1394 125 16 , , PUNCT ap-1394 125 17 σy σy PROPN ap-1394 125 18 , , PUNCT ap-1394 125 19 σz σz PRON ap-1394 125 20 defined define VERB ap-1394 125 21 by by ADP ap-1394 125 22 ( ( PUNCT ap-1394 125 23 18 18 NUM ap-1394 125 24 ) ) PUNCT ap-1394 125 25 no no ADV ap-1394 125 26 longer long ADV ap-1394 125 27 obey obey VERB ap-1394 125 28 the the DET ap-1394 125 29 standard standard ADJ ap-1394 125 30 bloch bloch PROPN ap-1394 125 31 equations equation NOUN ap-1394 125 32 ( ( PUNCT ap-1394 125 33 19 19 NUM ap-1394 125 34 ) ) PUNCT ap-1394 125 35 ; ; PUNCT ap-1394 125 36 instead instead ADV ap-1394 125 37 , , PUNCT ap-1394 125 38 they they PRON ap-1394 125 39 satisfy satisfy VERB ap-1394 125 40 1 1 NUM ap-1394 125 41 2 2 NUM ap-1394 125 42 σ̇x σ̇x NOUN ap-1394 125 43 = = SYM ap-1394 125 44 −νσz −νσz NOUN ap-1394 125 45 − − NOUN ap-1394 126 1 u3σy u3σy PUNCT ap-1394 126 2 1 1 NUM ap-1394 126 3 2 2 NUM ap-1394 126 4 σ̇y σ̇y NOUN ap-1394 126 5 = = SYM ap-1394 127 1 −u3σx −u3σx NUM ap-1394 127 2 + + NOUN ap-1394 127 3 u1σz u1σz PUNCT ap-1394 127 4 ( ( PUNCT ap-1394 127 5 26 26 NUM ap-1394 127 6 ) ) SYM ap-1394 127 7 1 1 NUM ap-1394 127 8 2 2 NUM ap-1394 127 9 σ̇z σ̇z NOUN ap-1394 127 10 = = SYM ap-1394 127 11 u1σy u1σy PUNCT ap-1394 127 12 + + CCONJ ap-1394 127 13 νσx νσx PROPN ap-1394 127 14 , , PUNCT ap-1394 127 15 and and CCONJ ap-1394 127 16 preserve preserve VERB ap-1394 127 17 the the DET ap-1394 127 18 relation relation NOUN ap-1394 127 19 σ2x σ2x NOUN ap-1394 127 20 − − VERB ap-1394 127 21 σ2y σ2y NOUN ap-1394 127 22 + + CCONJ ap-1394 127 23 σ2z σ2z X ap-1394 127 24 = = SYM ap-1394 127 25 const const NOUN ap-1394 127 26 . . PUNCT ap-1394 128 1 ( ( PUNCT ap-1394 128 2 27 27 NUM ap-1394 128 3 ) ) PUNCT ap-1394 128 4 we we PRON ap-1394 128 5 thus thus ADV ap-1394 128 6 see see VERB ap-1394 128 7 that that SCONJ ap-1394 128 8 at at ADP ap-1394 128 9 the the DET ap-1394 128 10 level level NOUN ap-1394 128 11 of of ADP ap-1394 128 12 reduced reduce VERB ap-1394 128 13 spin spin NOUN ap-1394 128 14 variables variable NOUN ap-1394 128 15 in in ADP ap-1394 128 16 three three NUM ap-1394 128 17 dimensions dimension NOUN ap-1394 128 18 , , PUNCT ap-1394 128 19 the the DET ap-1394 128 20 state state NOUN ap-1394 128 21 space space NOUN ap-1394 128 22 changes change NOUN ap-1394 128 23 from from ADP ap-1394 128 24 a a DET ap-1394 128 25 two two NUM ap-1394 128 26 - - PUNCT ap-1394 128 27 sphere sphere NOUN ap-1394 128 28 ( ( PUNCT ap-1394 128 29 20 20 NUM ap-1394 128 30 ) ) PUNCT ap-1394 128 31 to to ADP ap-1394 128 32 a a DET ap-1394 128 33 hyperboloid hyperboloid NOUN ap-1394 128 34 ( ( PUNCT ap-1394 128 35 27 27 NUM ap-1394 128 36 ) ) PUNCT ap-1394 128 37 , , PUNCT ap-1394 128 38 as as SCONJ ap-1394 128 39 the the DET ap-1394 128 40 parameters parameter NOUN ap-1394 128 41 u2 u2 PROPN ap-1394 128 42 , , PUNCT ap-1394 128 43 u4 u4 PROPN ap-1394 128 44 , , PUNCT ap-1394 128 45 u5 u5 PROPN ap-1394 128 46 appearing appear VERB ap-1394 128 47 in in ADP ap-1394 128 48 the the DET ap-1394 128 49 hamiltonian hamiltonian ADJ ap-1394 128 50 17 17 NUM ap-1394 128 51 acta acta PROPN ap-1394 128 52 polytechnica polytechnica PROPN ap-1394 128 53 vol vol NOUN ap-1394 128 54 . . PUNCT ap-1394 129 1 51 51 NUM ap-1394 129 2 no no NOUN ap-1394 129 3 . . PUNCT ap-1394 130 1 4/2011 4/2011 NUM ap-1394 130 2 fig fig NOUN ap-1394 130 3 . . PUNCT ap-1394 131 1 2 2 NUM ap-1394 131 2 : : PUNCT ap-1394 131 3 ( ( PUNCT ap-1394 131 4 colour colour NOUN ap-1394 131 5 online online NOUN ap-1394 131 6 ) ) PUNCT ap-1394 131 7 conic conic ADJ ap-1394 131 8 sections section NOUN ap-1394 131 9 and and CCONJ ap-1394 131 10 pt pt NOUN ap-1394 131 11 phase phase NOUN ap-1394 131 12 transition transition NOUN ap-1394 131 13 : : PUNCT ap-1394 131 14 changes change NOUN ap-1394 131 15 of of ADP ap-1394 131 16 orbit orbit NOUN ap-1394 131 17 structures structure NOUN ap-1394 131 18 . . PUNCT ap-1394 132 1 a a DET ap-1394 132 2 projection projection NOUN ap-1394 132 3 of of ADP ap-1394 132 4 the the DET ap-1394 132 5 orbits orbit NOUN ap-1394 132 6 on on ADP ap-1394 132 7 the the DET ap-1394 132 8 hyperboloid hyperboloid NOUN ap-1394 132 9 , , PUNCT ap-1394 132 10 for for ADP ap-1394 132 11 parameters parameter NOUN ap-1394 132 12 just just ADV ap-1394 132 13 above above ADP ap-1394 132 14 the the DET ap-1394 132 15 transition transition NOUN ap-1394 132 16 to to ADP ap-1394 132 17 complex complex ADJ ap-1394 132 18 energy energy NOUN ap-1394 132 19 eigenvalues eigenvalue NOUN ap-1394 132 20 , , PUNCT ap-1394 132 21 is be AUX ap-1394 132 22 shown show VERB ap-1394 132 23 on on ADP ap-1394 132 24 the the DET ap-1394 132 25 left left ADJ ap-1394 132 26 side side NOUN ap-1394 132 27 . . PUNCT ap-1394 133 1 the the DET ap-1394 133 2 orbits orbit NOUN ap-1394 133 3 form form VERB ap-1394 133 4 circular circular ADJ ap-1394 133 5 sections section NOUN ap-1394 133 6 . . PUNCT ap-1394 134 1 on on ADP ap-1394 134 2 the the DET ap-1394 134 3 right right ADJ ap-1394 134 4 side side NOUN ap-1394 134 5 we we PRON ap-1394 134 6 plot plot VERB ap-1394 134 7 orbits orbit NOUN ap-1394 134 8 of of ADP ap-1394 134 9 hyperbolic hyperbolic ADJ ap-1394 134 10 rabi rabi NOUN ap-1394 134 11 oscillations oscillation NOUN ap-1394 134 12 further far ADV ap-1394 134 13 into into ADP ap-1394 134 14 complex complex ADJ ap-1394 134 15 energy energy NOUN ap-1394 134 16 eigenvalues eigenvalue NOUN ap-1394 134 17 . . PUNCT ap-1394 135 1 the the DET ap-1394 135 2 energy energy NOUN ap-1394 135 3 eigenvalues eigenvalue NOUN ap-1394 135 4 determine determine VERB ap-1394 135 5 the the DET ap-1394 135 6 angle angle NOUN ap-1394 135 7 between between ADP ap-1394 135 8 the the DET ap-1394 135 9 axis axis NOUN ap-1394 135 10 of of ADP ap-1394 135 11 rotation rotation NOUN ap-1394 135 12 and and CCONJ ap-1394 135 13 the the DET ap-1394 135 14 axis axis NOUN ap-1394 135 15 of of ADP ap-1394 135 16 the the DET ap-1394 135 17 hyperboloid hyperboloid NOUN ap-1394 135 18 . . PUNCT ap-1394 136 1 when when SCONJ ap-1394 136 2 eigenvalues eigenvalue NOUN ap-1394 136 3 are be AUX ap-1394 136 4 complex complex ADJ ap-1394 136 5 , , PUNCT ap-1394 136 6 the the DET ap-1394 136 7 axis axis NOUN ap-1394 136 8 of of ADP ap-1394 136 9 rotation rotation NOUN ap-1394 136 10 is be AUX ap-1394 136 11 within within ADP ap-1394 136 12 the the DET ap-1394 136 13 hyperboloid hyperboloid NOUN ap-1394 136 14 , , PUNCT ap-1394 136 15 leading lead VERB ap-1394 136 16 to to ADP ap-1394 136 17 closed closed ADJ ap-1394 136 18 orbits orbit NOUN ap-1394 136 19 on on ADP ap-1394 136 20 the the DET ap-1394 136 21 state state NOUN ap-1394 136 22 space space NOUN ap-1394 136 23 generated generate VERB ap-1394 136 24 by by ADP ap-1394 136 25 circular circular ADJ ap-1394 136 26 sections section NOUN ap-1394 136 27 . . PUNCT ap-1394 137 1 when when SCONJ ap-1394 137 2 the the DET ap-1394 137 3 imaginary imaginary ADJ ap-1394 137 4 part part NOUN ap-1394 137 5 of of ADP ap-1394 137 6 the the DET ap-1394 137 7 coquaternion coquaternion NOUN ap-1394 137 8 appearing appear VERB ap-1394 137 9 in in ADP ap-1394 137 10 the the DET ap-1394 137 11 hamiltonian hamiltonian NOUN ap-1394 137 12 is be AUX ap-1394 137 13 null null ADJ ap-1394 137 14 , , PUNCT ap-1394 137 15 we we PRON ap-1394 137 16 have have VERB ap-1394 137 17 parabolic parabolic ADJ ap-1394 137 18 sections section NOUN ap-1394 137 19 of of ADP ap-1394 137 20 the the DET ap-1394 137 21 hyperboloid hyperboloid NOUN ap-1394 137 22 ; ; PUNCT ap-1394 137 23 whereas whereas SCONJ ap-1394 137 24 when when SCONJ ap-1394 137 25 the the DET ap-1394 137 26 energy energy NOUN ap-1394 137 27 eigenvalues eigenvalue NOUN ap-1394 137 28 are be AUX ap-1394 137 29 real real ADJ ap-1394 137 30 , , PUNCT ap-1394 137 31 the the DET ap-1394 137 32 angle angle NOUN ap-1394 137 33 of of ADP ap-1394 137 34 the the DET ap-1394 137 35 two two NUM ap-1394 137 36 axes axis NOUN ap-1394 137 37 is be AUX ap-1394 137 38 larger large ADJ ap-1394 137 39 than than ADP ap-1394 137 40 π/4 π/4 NUM ap-1394 137 41 , , PUNCT ap-1394 137 42 and and CCONJ ap-1394 137 43 open open ADJ ap-1394 137 44 orbits orbit NOUN ap-1394 137 45 are be AUX ap-1394 137 46 generated generate VERB ap-1394 137 47 by by ADP ap-1394 137 48 hyperbolic hyperbolic ADJ ap-1394 137 49 sections section NOUN ap-1394 137 50 change change VERB ap-1394 137 51 . . PUNCT ap-1394 138 1 this this DET ap-1394 138 2 transition transition NOUN ap-1394 138 3 corresponds correspond VERB ap-1394 138 4 to to ADP ap-1394 138 5 the the DET ap-1394 138 6 transition transition NOUN ap-1394 138 7 from from ADP ap-1394 138 8 a a DET ap-1394 138 9 complex complex ADJ ap-1394 138 10 hermitian hermitian ADJ ap-1394 138 11 hamiltonian hamiltonian NOUN ap-1394 138 12 into into ADP ap-1394 138 13 a a DET ap-1394 138 14 pt pt ADJ ap-1394 138 15 - - PUNCT ap-1394 138 16 symmetric symmetric ADJ ap-1394 138 17 non non ADJ ap-1394 138 18 - - ADJ ap-1394 138 19 hermitian hermitian ADJ ap-1394 138 20 hamiltonian hamiltonian NOUN ap-1394 138 21 . . PUNCT ap-1394 139 1 since since SCONJ ap-1394 139 2 the the DET ap-1394 139 3 energy energy NOUN ap-1394 139 4 eigenvalues eigenvalue NOUN ap-1394 139 5 can can AUX ap-1394 139 6 still still ADV ap-1394 139 7 be be AUX ap-1394 139 8 real real ADJ ap-1394 139 9 even even ADV ap-1394 139 10 when when SCONJ ap-1394 139 11 u22 u22 PROPN ap-1394 139 12 < < X ap-1394 139 13 u24 u24 PROPN ap-1394 139 14 + + CCONJ ap-1394 139 15 u25 u25 VERB ap-1394 139 16 , , PUNCT ap-1394 139 17 we we PRON ap-1394 139 18 expect expect VERB ap-1394 139 19 the the DET ap-1394 139 20 dynamics dynamic NOUN ap-1394 139 21 to to PART ap-1394 139 22 exhibit exhibit VERB ap-1394 139 23 two two NUM ap-1394 139 24 distinct distinct ADJ ap-1394 139 25 characteristics characteristic NOUN ap-1394 139 26 depending depend VERB ap-1394 139 27 on on ADP ap-1394 139 28 whether whether SCONJ ap-1394 139 29 the the DET ap-1394 139 30 reality reality NOUN ap-1394 139 31 condition condition NOUN ap-1394 139 32 u21 u21 NOUN ap-1394 139 33 + + CCONJ ap-1394 139 34 u22 u22 PROPN ap-1394 139 35 + + CCONJ ap-1394 139 36 u23 u23 PROPN ap-1394 139 37 > > X ap-1394 139 38 u24 u24 PROPN ap-1394 139 39 + + CCONJ ap-1394 139 40 u25 u25 PRON ap-1394 139 41 is be AUX ap-1394 139 42 satisfied satisfied ADJ ap-1394 139 43 . . PUNCT ap-1394 140 1 indeed indeed ADV ap-1394 140 2 , , PUNCT ap-1394 140 3 we we PRON ap-1394 140 4 find find VERB ap-1394 140 5 that that SCONJ ap-1394 140 6 on on ADP ap-1394 140 7 a a DET ap-1394 140 8 hyperbolic hyperbolic ADJ ap-1394 140 9 state state NOUN ap-1394 140 10 space space NOUN ap-1394 140 11 , , PUNCT ap-1394 140 12 orbits orbit NOUN ap-1394 140 13 of of ADP ap-1394 140 14 the the DET ap-1394 140 15 unitary unitary ADJ ap-1394 140 16 dynamics dynamic NOUN ap-1394 140 17 associated associate VERB ap-1394 140 18 with with ADP ap-1394 140 19 real real ADJ ap-1394 140 20 energies energy NOUN ap-1394 140 21 are be AUX ap-1394 140 22 the the DET ap-1394 140 23 ones one NOUN ap-1394 140 24 that that PRON ap-1394 140 25 are be AUX ap-1394 140 26 open open ADJ ap-1394 140 27 and and CCONJ ap-1394 140 28 run run VERB ap-1394 140 29 off off ADP ap-1394 140 30 to to ADP ap-1394 140 31 infinities infinity NOUN ap-1394 140 32 . . PUNCT ap-1394 141 1 conversely conversely ADV ap-1394 141 2 , , PUNCT ap-1394 141 3 when when SCONJ ap-1394 141 4 the the DET ap-1394 141 5 reality reality NOUN ap-1394 141 6 condition condition NOUN ap-1394 141 7 is be AUX ap-1394 141 8 violated violate VERB ap-1394 141 9 , , PUNCT ap-1394 141 10 these these DET ap-1394 141 11 open open ADJ ap-1394 141 12 orbits orbit NOUN ap-1394 141 13 are be AUX ap-1394 141 14 in in ADP ap-1394 141 15 effect effect NOUN ap-1394 141 16 wick wick NOUN ap-1394 141 17 rotated rotate VERB ap-1394 141 18 to to PART ap-1394 141 19 generate generate VERB ap-1394 141 20 closed closed ADJ ap-1394 141 21 orbits orbit NOUN ap-1394 141 22 . . PUNCT ap-1394 142 1 these these DET ap-1394 142 2 features feature NOUN ap-1394 142 3 can can AUX ap-1394 142 4 be be AUX ap-1394 142 5 identified identify VERB ap-1394 142 6 by by ADP ap-1394 142 7 a a DET ap-1394 142 8 closer close ADJ ap-1394 142 9 inspection inspection NOUN ap-1394 142 10 on on ADP ap-1394 142 11 the the DET ap-1394 142 12 structure structure NOUN ap-1394 142 13 of of ADP ap-1394 142 14 the the DET ap-1394 142 15 underlying underlie VERB ap-1394 142 16 state state NOUN ap-1394 142 17 space space NOUN ap-1394 142 18 , , PUNCT ap-1394 142 19 upon upon SCONJ ap-1394 142 20 which which PRON ap-1394 142 21 the the DET ap-1394 142 22 dynamical dynamical ADJ ap-1394 142 23 orbits orbit NOUN ap-1394 142 24 lie lie VERB ap-1394 142 25 . . PUNCT ap-1394 143 1 in in ADP ap-1394 143 2 particular particular ADJ ap-1394 143 3 , , PUNCT ap-1394 143 4 ( ( PUNCT ap-1394 143 5 26 26 NUM ap-1394 143 6 ) ) PUNCT ap-1394 143 7 shows show VERB ap-1394 143 8 that that SCONJ ap-1394 143 9 the the DET ap-1394 143 10 dynamics dynamic NOUN ap-1394 143 11 generates generate VERB ap-1394 143 12 a a DET ap-1394 143 13 rotation rotation NOUN ap-1394 143 14 around around ADP ap-1394 143 15 the the DET ap-1394 143 16 axis axis NOUN ap-1394 143 17 ( ( PUNCT ap-1394 143 18 u1 u1 NOUN ap-1394 143 19 , , PUNCT ap-1394 143 20 ν ν NOUN ap-1394 143 21 , , PUNCT ap-1394 143 22 u3 u3 PROPN ap-1394 143 23 ) ) PUNCT ap-1394 143 24 ; ; PUNCT ap-1394 143 25 whereas whereas SCONJ ap-1394 143 26 the the DET ap-1394 143 27 state state NOUN ap-1394 143 28 space space NOUN ap-1394 143 29 ( ( PUNCT ap-1394 143 30 27 27 NUM ap-1394 143 31 ) ) PUNCT ap-1394 143 32 is be AUX ap-1394 143 33 a a DET ap-1394 143 34 hyperboloid hyperboloid NOUN ap-1394 143 35 about about ADP ap-1394 143 36 the the DET ap-1394 143 37 axis axis NOUN ap-1394 143 38 ( ( PUNCT ap-1394 143 39 0 0 NUM ap-1394 143 40 , , PUNCT ap-1394 143 41 1 1 NUM ap-1394 143 42 , , PUNCT ap-1394 143 43 0 0 NUM ap-1394 143 44 ) ) PUNCT ap-1394 143 45 . . PUNCT ap-1394 144 1 we we PRON ap-1394 144 2 have have AUX ap-1394 144 3 sketched sketch VERB ap-1394 144 4 in in ADP ap-1394 144 5 figure figure NOUN ap-1394 144 6 2 2 NUM ap-1394 144 7 dynamical dynamical ADJ ap-1394 144 8 orbits orbit NOUN ap-1394 144 9 resulting result VERB ap-1394 144 10 from from ADP ap-1394 144 11 ( ( PUNCT ap-1394 144 12 26 26 NUM ap-1394 144 13 ) ) PUNCT ap-1394 144 14 , , PUNCT ap-1394 144 15 indicating indicate VERB ap-1394 144 16 that that SCONJ ap-1394 144 17 there there PRON ap-1394 144 18 indeed indeed ADV ap-1394 144 19 is be AUX ap-1394 144 20 a a DET ap-1394 144 21 transition transition NOUN ap-1394 144 22 from from ADP ap-1394 144 23 open open ADJ ap-1394 144 24 to to ADP ap-1394 144 25 closed closed ADJ ap-1394 144 26 orbits orbit NOUN ap-1394 144 27 as as SCONJ ap-1394 144 28 real real ADJ ap-1394 144 29 eigenvalues eigenvalue NOUN ap-1394 144 30 turn turn VERB ap-1394 144 31 into into ADP ap-1394 144 32 complex complex ADJ ap-1394 144 33 conjugate conjugate ADJ ap-1394 144 34 pairs pair NOUN ap-1394 144 35 . . PUNCT ap-1394 145 1 intuitively intuitively ADV ap-1394 145 2 , , PUNCT ap-1394 145 3 one one PRON ap-1394 145 4 might might AUX ap-1394 145 5 have have AUX ap-1394 145 6 expected expect VERB ap-1394 145 7 an an DET ap-1394 145 8 opposite opposite ADJ ap-1394 145 9 transition transition NOUN ap-1394 145 10 since since SCONJ ap-1394 145 11 in in ADP ap-1394 145 12 a a DET ap-1394 145 13 pt pt ADJ ap-1394 145 14 - - PUNCT ap-1394 145 15 symmetric symmetric ADJ ap-1394 145 16 model model NOUN ap-1394 145 17 of of ADP ap-1394 145 18 a a DET ap-1394 145 19 spin12 spin12 ADJ ap-1394 145 20 system system NOUN ap-1394 145 21 the the DET ap-1394 145 22 renormalised renormalise VERB ap-1394 145 23 bloch bloch NOUN ap-1394 145 24 vectors vector NOUN ap-1394 145 25 on on ADP ap-1394 145 26 a a DET ap-1394 145 27 spherical spherical ADJ ap-1394 145 28 state state NOUN ap-1394 145 29 space space NOUN ap-1394 145 30 follow follow NOUN ap-1394 145 31 closed close VERB ap-1394 145 32 orbits orbit NOUN ap-1394 145 33 when when SCONJ ap-1394 145 34 eigenvalues eigenvalue NOUN ap-1394 145 35 are be AUX ap-1394 145 36 real real ADJ ap-1394 145 37 , , PUNCT ap-1394 145 38 whereas whereas SCONJ ap-1394 145 39 sinks sink NOUN ap-1394 145 40 and and CCONJ ap-1394 145 41 sources source NOUN ap-1394 145 42 are be AUX ap-1394 145 43 created create VERB ap-1394 145 44 when when SCONJ ap-1394 145 45 eigenvalues eigenvalue NOUN ap-1394 145 46 become become VERB ap-1394 145 47 complex complex ADJ ap-1394 146 1 [ [ X ap-1394 146 2 15 15 NUM ap-1394 146 3 ] ] PUNCT ap-1394 146 4 . . PUNCT ap-1394 147 1 the the DET ap-1394 147 2 apparent apparent ADJ ap-1394 147 3 opposite opposite ADJ ap-1394 147 4 behaviour behaviour NOUN ap-1394 147 5 seen see VERB ap-1394 147 6 here here ADV ap-1394 147 7 is be AUX ap-1394 147 8 presumably presumably ADV ap-1394 147 9 to to PART ap-1394 147 10 do do VERB ap-1394 147 11 with with ADP ap-1394 147 12 the the DET ap-1394 147 13 facts fact NOUN ap-1394 147 14 that that SCONJ ap-1394 147 15 the the DET ap-1394 147 16 underlying underlie VERB ap-1394 147 17 state state NOUN ap-1394 147 18 space space NOUN ap-1394 147 19 is be AUX ap-1394 147 20 hyperbolic hyperbolic ADJ ap-1394 147 21 , , PUNCT ap-1394 147 22 not not PART ap-1394 147 23 spherical spherical ADJ ap-1394 147 24 , , PUNCT ap-1394 147 25 and and CCONJ ap-1394 147 26 that that SCONJ ap-1394 147 27 no no DET ap-1394 147 28 renormalisation renormalisation NOUN ap-1394 147 29 is be AUX ap-1394 147 30 performed perform VERB ap-1394 147 31 here here ADV ap-1394 147 32 . . PUNCT ap-1394 148 1 in in ADP ap-1394 148 2 figure figure NOUN ap-1394 148 3 2 2 NUM ap-1394 148 4 we we PRON ap-1394 148 5 have have AUX ap-1394 148 6 sketched sketch VERB ap-1394 148 7 some some DET ap-1394 148 8 dynamical dynamical ADJ ap-1394 148 9 trajectories trajectory NOUN ap-1394 148 10 when when SCONJ ap-1394 148 11 energy energy NOUN ap-1394 148 12 eigenvalues eigenvalue NOUN ap-1394 148 13 are be AUX ap-1394 148 14 complex complex ADJ ap-1394 148 15 . . PUNCT ap-1394 149 1 a a DET ap-1394 149 2 projection projection NOUN ap-1394 149 3 of of ADP ap-1394 149 4 the the DET ap-1394 149 5 dynamical dynamical ADJ ap-1394 149 6 orbits orbit NOUN ap-1394 149 7 from from ADP ap-1394 149 8 the the DET ap-1394 149 9 σz σz NOUN ap-1394 149 10 axis axis NOUN ap-1394 149 11 ( ( PUNCT ap-1394 149 12 for for ADP ap-1394 149 13 the the DET ap-1394 149 14 choice choice NOUN ap-1394 149 15 of of ADP ap-1394 149 16 parameters parameter NOUN ap-1394 149 17 used use VERB ap-1394 149 18 in in ADP ap-1394 149 19 these these DET ap-1394 149 20 plots plot NOUN ap-1394 149 21 ) ) PUNCT ap-1394 149 22 shows show NOUN ap-1394 149 23 in in ADP ap-1394 149 24 which which DET ap-1394 149 25 way way NOUN ap-1394 149 26 the the DET ap-1394 149 27 topology topology NOUN ap-1394 149 28 of of ADP ap-1394 149 29 the the DET ap-1394 149 30 orbits orbit NOUN ap-1394 149 31 are be AUX ap-1394 149 32 affected affect VERB ap-1394 149 33 by by ADP ap-1394 149 34 the the DET ap-1394 149 35 reality reality NOUN ap-1394 149 36 of of ADP ap-1394 149 37 the the DET ap-1394 149 38 energy energy NOUN ap-1394 149 39 eigenvalues eigenvalue VERB ap-1394 149 40 . . PUNCT ap-1394 150 1 the the DET ap-1394 150 2 evolutions evolution NOUN ap-1394 150 3 of of ADP ap-1394 150 4 the the DET ap-1394 150 5 other other ADJ ap-1394 150 6 dynamical dynamical ADJ ap-1394 150 7 variables variable NOUN ap-1394 150 8 σ2 σ2 PROPN ap-1394 150 9 , , PUNCT ap-1394 150 10 σ4 σ4 NOUN ap-1394 150 11 , , PUNCT ap-1394 150 12 σ5 σ5 NOUN ap-1394 150 13 are be AUX ap-1394 150 14 confined confine VERB ap-1394 150 15 to to ADP ap-1394 150 16 the the DET ap-1394 150 17 space space NOUN ap-1394 150 18 characterised characterise VERB ap-1394 150 19 by by ADP ap-1394 150 20 the the DET ap-1394 150 21 relation relation NOUN ap-1394 150 22 ( ( PUNCT ap-1394 150 23 u2σ4 u2σ4 ADP ap-1394 150 24 + + NUM ap-1394 150 25 u4σ2)2 u4σ2)2 NOUN ap-1394 150 26 − − PRON ap-1394 151 1 ( ( PUNCT ap-1394 151 2 u4σ5 u4σ5 INTJ ap-1394 151 3 − − PROPN ap-1394 152 1 u5σ4)2 u5σ4)2 ADJ ap-1394 152 2 + + ADJ ap-1394 152 3 ( ( PUNCT ap-1394 152 4 u5σ2 u5σ2 X ap-1394 152 5 + + NUM ap-1394 152 6 u2σ5 u2σ5 NOUN ap-1394 152 7 ) ) PUNCT ap-1394 152 8 2 2 NUM ap-1394 152 9 = = SYM ap-1394 152 10 const const NOUN ap-1394 152 11 . . PUNCT ap-1394 152 12 , , PUNCT ap-1394 152 13 ( ( PUNCT ap-1394 152 14 28 28 NUM ap-1394 152 15 ) ) PUNCT ap-1394 152 16 instead instead ADV ap-1394 152 17 of of ADP ap-1394 152 18 the the DET ap-1394 152 19 relation relation NOUN ap-1394 152 20 ( ( PUNCT ap-1394 152 21 21 21 NUM ap-1394 152 22 ) ) PUNCT ap-1394 152 23 of of ADP ap-1394 152 24 the the DET ap-1394 152 25 previous previous ADJ ap-1394 152 26 case case NOUN ap-1394 152 27 . . PUNCT ap-1394 153 1 however however ADV ap-1394 153 2 , , PUNCT ap-1394 153 3 if if SCONJ ap-1394 153 4 we we PRON ap-1394 153 5 define define VERB ap-1394 153 6 , , PUNCT ap-1394 153 7 as as ADP ap-1394 153 8 before before ADV ap-1394 153 9 , , PUNCT ap-1394 153 10 three three NUM ap-1394 153 11 auxiliary auxiliary ADJ ap-1394 153 12 variables variable NOUN ap-1394 153 13 σy1 σy1 PROPN ap-1394 154 1 = = SYM ap-1394 154 2 u4σ5 u4σ5 PROPN ap-1394 154 3 − − PROPN ap-1394 154 4 u5σ4 u5σ4 NOUN ap-1394 154 5 , , PUNCT ap-1394 154 6 σy2 σy2 NOUN ap-1394 154 7 = = SYM ap-1394 154 8 u5σ2 u5σ2 X ap-1394 154 9 + + CCONJ ap-1394 154 10 u2σ5 u2σ5 NOUN ap-1394 154 11 , , PUNCT ap-1394 154 12 and and CCONJ ap-1394 154 13 σy3 σy3 PROPN ap-1394 154 14 = = SYM ap-1394 155 1 u2σ4 u2σ4 PROPN ap-1394 155 2 + + X ap-1394 155 3 u4σ2 u4σ2 ADJ ap-1394 155 4 , , PUNCT ap-1394 155 5 then then ADV ap-1394 155 6 the the DET ap-1394 155 7 dynamical dynamical ADJ ap-1394 155 8 equations equation NOUN ap-1394 155 9 satisfied satisfy VERB ap-1394 155 10 by by ADP ap-1394 155 11 these these DET ap-1394 155 12 variables variable NOUN ap-1394 155 13 are be AUX ap-1394 155 14 identical identical ADJ ap-1394 155 15 to to ADP ap-1394 155 16 those those PRON ap-1394 155 17 in in ADP ap-1394 155 18 ( ( PUNCT ap-1394 155 19 22 22 NUM ap-1394 155 20 ) ) PUNCT ap-1394 155 21 , , PUNCT ap-1394 155 22 except except SCONJ ap-1394 155 23 , , PUNCT ap-1394 155 24 of of ADP ap-1394 155 25 course course NOUN ap-1394 155 26 , , PUNCT ap-1394 155 27 that that SCONJ ap-1394 155 28 the the DET ap-1394 155 29 definition definition NOUN ap-1394 155 30 of of ADP ap-1394 155 31 ν ν NOUN ap-1394 155 32 is be AUX ap-1394 155 33 different different ADJ ap-1394 155 34 . . PUNCT ap-1394 156 1 it it PRON ap-1394 156 2 is be AUX ap-1394 156 3 interesting interesting ADJ ap-1394 156 4 to to PART ap-1394 156 5 remark remark VERB ap-1394 156 6 that that SCONJ ap-1394 156 7 when when SCONJ ap-1394 156 8 the the DET ap-1394 156 9 imaginary imaginary ADJ ap-1394 156 10 part part NOUN ap-1394 156 11 of of ADP ap-1394 156 12 the the DET ap-1394 156 13 coquaternion coquaternion NOUN ap-1394 156 14 appearing appear VERB ap-1394 156 15 in in ADP ap-1394 156 16 the the DET ap-1394 156 17 hamiltonian hamiltonian NOUN ap-1394 156 18 is be AUX ap-1394 156 19 space space NOUN ap-1394 156 20 - - PUNCT ap-1394 156 21 like like ADJ ap-1394 156 22 , , PUNCT ap-1394 156 23 the the DET ap-1394 156 24 imaginary imaginary ADJ ap-1394 156 25 unit unit NOUN ap-1394 156 26 i i PRON ap-1394 156 27 has have VERB ap-1394 156 28 the the DET ap-1394 156 29 characteristic characteristic NOUN ap-1394 156 30 of of ADP ap-1394 156 31 a a DET ap-1394 156 32 ‘ ' PUNCT ap-1394 156 33 double double ADJ ap-1394 156 34 number number NOUN ap-1394 156 35 ’ ' PUNCT ap-1394 156 36 or or CCONJ ap-1394 156 37 a a DET ap-1394 156 38 ‘ ' PUNCT ap-1394 156 39 study study NOUN ap-1394 156 40 number number NOUN ap-1394 156 41 ’ ' PUNCT ap-1394 156 42 introduced introduce VERB ap-1394 156 43 by by ADP ap-1394 156 44 clifford clifford PROPN ap-1394 157 1 [ [ X ap-1394 157 2 26 26 NUM ap-1394 157 3 ] ] X ap-1394 157 4 , , PUNCT ap-1394 157 5 that that ADV ap-1394 157 6 is is ADV ap-1394 157 7 , , PUNCT ap-1394 157 8 i2 i2 PROPN ap-1394 157 9 = = SYM ap-1394 157 10 1 1 X ap-1394 157 11 . . PUNCT ap-1394 157 12 quantum quantum NOUN ap-1394 157 13 theories theory NOUN ap-1394 157 14 generated generate VERB ap-1394 157 15 by by ADP ap-1394 157 16 such such DET ap-1394 157 17 a a DET ap-1394 157 18 number number NOUN ap-1394 157 19 field field NOUN ap-1394 157 20 ( ( PUNCT ap-1394 157 21 instead instead ADV ap-1394 157 22 of of ADP ap-1394 157 23 the the DET ap-1394 157 24 field field NOUN ap-1394 157 25 of of ADP ap-1394 157 26 complex complex ADJ ap-1394 157 27 numbers number NOUN ap-1394 157 28 ) ) PUNCT ap-1394 157 29 and and CCONJ ap-1394 157 30 other other ADJ ap-1394 157 31 hyperbolic hyperbolic ADJ ap-1394 157 32 generalisations generalisation NOUN ap-1394 157 33 , , PUNCT ap-1394 157 34 as as ADV ap-1394 157 35 well well ADV ap-1394 157 36 as as ADP ap-1394 157 37 various various ADJ ap-1394 157 38 issues issue NOUN ap-1394 157 39 that that PRON ap-1394 157 40 might might AUX ap-1394 157 41 arise arise VERB ap-1394 157 42 from from ADP ap-1394 157 43 such such ADJ ap-1394 157 44 generalisations generalisation NOUN ap-1394 157 45 , , PUNCT ap-1394 157 46 have have AUX ap-1394 157 47 been be AUX ap-1394 157 48 discussed discuss VERB ap-1394 157 49 by by ADP ap-1394 157 50 various various ADJ ap-1394 157 51 authors author NOUN ap-1394 157 52 ( ( PUNCT ap-1394 157 53 e.g. e.g. ADV ap-1394 157 54 , , PUNCT ap-1394 157 55 [ [ X ap-1394 157 56 27 27 NUM ap-1394 157 57 , , PUNCT ap-1394 157 58 28 28 NUM ap-1394 157 59 ] ] PUNCT ap-1394 157 60 ; ; PUNCT ap-1394 157 61 see see VERB ap-1394 157 62 also also ADV ap-1394 157 63 [ [ X ap-1394 157 64 23 23 NUM ap-1394 157 65 ] ] PUNCT ap-1394 157 66 and and CCONJ ap-1394 157 67 references reference NOUN ap-1394 157 68 cited cite VERB ap-1394 157 69 therein therein ADV ap-1394 157 70 ) ) PUNCT ap-1394 157 71 . . PUNCT ap-1394 158 1 the the DET ap-1394 158 2 use use NOUN ap-1394 158 3 of of ADP ap-1394 158 4 coquaternionic coquaternionic ADJ ap-1394 158 5 hermitian hermitian ADJ ap-1394 158 6 hamiltonian hamiltonian NOUN ap-1394 158 7 thus thus ADV ap-1394 158 8 captures capture VERB ap-1394 158 9 dynamical dynamical ADJ ap-1394 158 10 behaviours behaviour NOUN ap-1394 158 11 of of ADP ap-1394 158 12 different different ADJ ap-1394 158 13 generalisations generalisation NOUN ap-1394 158 14 of of ADP ap-1394 158 15 quantum quantum ADJ ap-1394 158 16 mechanics mechanic NOUN ap-1394 158 17 in in ADP ap-1394 158 18 a a DET ap-1394 158 19 simple simple ADJ ap-1394 158 20 unified unified ADJ ap-1394 158 21 scheme scheme NOUN ap-1394 158 22 . . PUNCT ap-1394 159 1 18 18 NUM ap-1394 159 2 acta acta PROPN ap-1394 159 3 polytechnica polytechnica PROPN ap-1394 159 4 vol vol NOUN ap-1394 159 5 . . PUNCT ap-1394 160 1 51 51 NUM ap-1394 160 2 no no INTJ ap-1394 160 3 . . PUNCT ap-1394 161 1 4/2011 4/2011 NUM ap-1394 161 2 acknowledgement acknowledgement NOUN ap-1394 161 3 we we PRON ap-1394 161 4 thank thank VERB ap-1394 161 5 the the DET ap-1394 161 6 participants participant NOUN ap-1394 161 7 of of ADP ap-1394 161 8 the the DET ap-1394 161 9 international international ADJ ap-1394 161 10 conference conference NOUN ap-1394 161 11 on on ADP ap-1394 161 12 analytic analytic ADJ ap-1394 161 13 and and CCONJ ap-1394 161 14 algebraic algebraic ADJ ap-1394 161 15 methods method NOUN ap-1394 161 16 in in ADP ap-1394 161 17 physics physics PROPN ap-1394 161 18 vii vii PROPN ap-1394 161 19 , , PUNCT ap-1394 161 20 prague prague PROPN ap-1394 161 21 , , PUNCT ap-1394 161 22 2011 2011 NUM ap-1394 161 23 , , PUNCT ap-1394 161 24 for for ADP ap-1394 161 25 stimulating stimulate VERB ap-1394 161 26 discussions discussion NOUN ap-1394 161 27 . . PUNCT ap-1394 162 1 emg emg NOUN ap-1394 162 2 is be AUX ap-1394 162 3 supported support VERB ap-1394 162 4 by by ADP ap-1394 162 5 an an DET ap-1394 162 6 imperial imperial ADJ ap-1394 162 7 college college NOUN ap-1394 162 8 junior junior ADJ ap-1394 162 9 research research NOUN ap-1394 162 10 fellowship fellowship NOUN ap-1394 162 11 . . PUNCT ap-1394 163 1 references reference NOUN ap-1394 163 2 [ [ X ap-1394 163 3 1 1 NUM ap-1394 163 4 ] ] X ap-1394 163 5 bender bender NOUN ap-1394 163 6 , , PUNCT ap-1394 163 7 c. c. PROPN ap-1394 163 8 m. m. PROPN ap-1394 163 9 , , PUNCT ap-1394 163 10 boettcher boettcher PROPN ap-1394 163 11 , , PUNCT ap-1394 163 12 s. s. PROPN ap-1394 163 13 , , PUNCT ap-1394 163 14 meisinger meisinger NOUN ap-1394 163 15 , , PUNCT ap-1394 163 16 p. p. PROPN ap-1394 163 17 n. n. NOUN ap-1394 163 18 : : PUNCT ap-1394 163 19 pt pt ADJ ap-1394 163 20 - - ADJ ap-1394 163 21 symmetric symmetric ADJ ap-1394 163 22 quantum quantum NOUN ap-1394 163 23 mechanics mechanic NOUN ap-1394 163 24 , , PUNCT ap-1394 163 25 j. j. PROPN ap-1394 163 26 math math PROPN ap-1394 163 27 . . PUNCT ap-1394 164 1 phys phy NOUN ap-1394 164 2 . . PUNCT ap-1394 165 1 40 40 NUM ap-1394 165 2 , , PUNCT ap-1394 165 3 1999 1999 NUM ap-1394 165 4 , , PUNCT ap-1394 165 5 2201 2201 NUM ap-1394 165 6 . . PUNCT ap-1394 166 1 [ [ X ap-1394 166 2 2 2 NUM ap-1394 166 3 ] ] X ap-1394 166 4 kaushal kaushal PROPN ap-1394 166 5 , , PUNCT ap-1394 166 6 r. r. PROPN ap-1394 166 7 s. s. PROPN ap-1394 166 8 , , PUNCT ap-1394 166 9 korsch korsch PROPN ap-1394 166 10 , , PUNCT ap-1394 166 11 h. h. PROPN ap-1394 166 12 j. j. PROPN ap-1394 166 13 : : PUNCT ap-1394 166 14 some some DET ap-1394 166 15 remarks remark NOUN ap-1394 166 16 on on ADP ap-1394 166 17 complex complex ADJ ap-1394 166 18 hamiltonian hamiltonian ADJ ap-1394 166 19 systems system NOUN ap-1394 166 20 , , PUNCT ap-1394 166 21 phys phy NOUN ap-1394 166 22 . . PUNCT ap-1394 167 1 lett lett PROPN ap-1394 167 2 . . PUNCT ap-1394 168 1 a a DET ap-1394 168 2 276 276 NUM ap-1394 168 3 , , PUNCT ap-1394 168 4 2000 2000 NUM ap-1394 168 5 , , PUNCT ap-1394 168 6 47 47 NUM ap-1394 168 7 . . PUNCT ap-1394 169 1 [ [ X ap-1394 169 2 3 3 NUM ap-1394 169 3 ] ] X ap-1394 169 4 mostafazadeh mostafazadeh NOUN ap-1394 169 5 , , PUNCT ap-1394 169 6 a. a. NOUN ap-1394 169 7 : : PUNCT ap-1394 169 8 real real ADJ ap-1394 169 9 description description NOUN ap-1394 169 10 of of ADP ap-1394 169 11 classical classical ADJ ap-1394 169 12 hamiltonian hamiltonian ADJ ap-1394 169 13 dynamics dynamic NOUN ap-1394 169 14 generated generate VERB ap-1394 169 15 by by ADP ap-1394 169 16 a a DET ap-1394 169 17 complex complex ADJ ap-1394 169 18 potential potential NOUN ap-1394 169 19 , , PUNCT ap-1394 169 20 phys phy NOUN ap-1394 169 21 . . PUNCT ap-1394 170 1 lett lett PROPN ap-1394 170 2 . . PUNCT ap-1394 171 1 a a DET ap-1394 171 2 357 357 NUM ap-1394 171 3 , , PUNCT ap-1394 171 4 2006 2006 NUM ap-1394 171 5 , , PUNCT ap-1394 171 6 177 177 NUM ap-1394 171 7 . . PUNCT ap-1394 172 1 [ [ X ap-1394 172 2 4 4 NUM ap-1394 172 3 ] ] X ap-1394 172 4 bender bender NOUN ap-1394 172 5 , , PUNCT ap-1394 172 6 c. c. PROPN ap-1394 172 7 m. m. PROPN ap-1394 172 8 , , PUNCT ap-1394 172 9 holm holm PROPN ap-1394 172 10 , , PUNCT ap-1394 172 11 d. d. PROPN ap-1394 172 12 d. d. PROPN ap-1394 172 13 , , PUNCT ap-1394 172 14 hook hook NOUN ap-1394 172 15 , , PUNCT ap-1394 172 16 d. d. PROPN ap-1394 172 17 w. w. PROPN ap-1394 172 18 : : PUNCT ap-1394 172 19 complex complex ADJ ap-1394 172 20 trajectories trajectory NOUN ap-1394 172 21 of of ADP ap-1394 172 22 a a DET ap-1394 172 23 simple simple ADJ ap-1394 172 24 pendulum pendulum NOUN ap-1394 172 25 , , PUNCT ap-1394 172 26 j. j. PROPN ap-1394 172 27 phys phys PROPN ap-1394 172 28 . . PUNCT ap-1394 173 1 a a DET ap-1394 173 2 40 40 NUM ap-1394 173 3 , , PUNCT ap-1394 173 4 2007 2007 NUM ap-1394 173 5 , , PUNCT ap-1394 173 6 f81 f81 NOUN ap-1394 173 7 . . PUNCT ap-1394 174 1 [ [ X ap-1394 174 2 5 5 NUM ap-1394 174 3 ] ] X ap-1394 174 4 bender bender NOUN ap-1394 174 5 , , PUNCT ap-1394 174 6 c. c. PROPN ap-1394 174 7 m. m. NOUN ap-1394 174 8 , , PUNCT ap-1394 174 9 hook hook NOUN ap-1394 174 10 , , PUNCT ap-1394 174 11 d. d. PROPN ap-1394 174 12 w. w. PROPN ap-1394 174 13 , , PUNCT ap-1394 174 14 kooner kooner PROPN ap-1394 174 15 , , PUNCT ap-1394 174 16 k. k. PROPN ap-1394 174 17 s. s. PROPN ap-1394 174 18 : : PUNCT ap-1394 174 19 classical classical ADJ ap-1394 174 20 particle particle NOUN ap-1394 174 21 in in ADP ap-1394 174 22 a a DET ap-1394 174 23 complex complex ADJ ap-1394 174 24 elliptic elliptic ADJ ap-1394 174 25 pendulum pendulum NOUN ap-1394 174 26 , , PUNCT ap-1394 174 27 j. j. PROPN ap-1394 174 28 phys phys PROPN ap-1394 174 29 . . PUNCT ap-1394 175 1 a a DET ap-1394 175 2 43 43 NUM ap-1394 175 3 , , PUNCT ap-1394 175 4 2010 2010 NUM ap-1394 175 5 , , PUNCT ap-1394 175 6 165 165 NUM ap-1394 175 7 201 201 NUM ap-1394 175 8 . . PUNCT ap-1394 176 1 [ [ X ap-1394 176 2 6 6 NUM ap-1394 176 3 ] ] SYM ap-1394 176 4 cavaglia cavaglia PROPN ap-1394 176 5 , , PUNCT ap-1394 176 6 a. a. NOUN ap-1394 176 7 , , PUNCT ap-1394 176 8 fring fring NOUN ap-1394 176 9 , , PUNCT ap-1394 176 10 a. a. NOUN ap-1394 176 11 , , PUNCT ap-1394 176 12 bagchi bagchi PROPN ap-1394 176 13 , , PUNCT ap-1394 176 14 b. b. PROPN ap-1394 176 15 : : PUNCT ap-1394 176 16 pt pt PROPN ap-1394 176 17 - - PUNCT ap-1394 176 18 symmetry symmetry NOUN ap-1394 176 19 breaking breaking NOUN ap-1394 176 20 in in ADP ap-1394 176 21 complex complex ADJ ap-1394 176 22 nonlinear nonlinear ADJ ap-1394 176 23 wave wave NOUN ap-1394 176 24 equations equation NOUN ap-1394 176 25 and and CCONJ ap-1394 176 26 their their PRON ap-1394 176 27 deformations deformation NOUN ap-1394 176 28 . . PUNCT ap-1394 177 1 j. j. PROPN ap-1394 177 2 phys phys PROPN ap-1394 177 3 . . PUNCT ap-1394 178 1 a a DET ap-1394 178 2 44 44 NUM ap-1394 178 3 , , PUNCT ap-1394 178 4 2011 2011 NUM ap-1394 178 5 , , PUNCT ap-1394 178 6 325201 325201 NUM ap-1394 178 7 , , PUNCT ap-1394 178 8 arxiv:1103.1832 arxiv:1103.1832 X ap-1394 178 9 [ [ X ap-1394 178 10 7 7 NUM ap-1394 178 11 ] ] X ap-1394 178 12 bender bender NOUN ap-1394 178 13 , , PUNCT ap-1394 178 14 c. c. PROPN ap-1394 178 15 m. m. NOUN ap-1394 178 16 , , PUNCT ap-1394 178 17 boettcher boettcher PROPN ap-1394 178 18 s. s. PROPN ap-1394 178 19 : : PUNCT ap-1394 178 20 real real ADJ ap-1394 178 21 spectra spectra NOUN ap-1394 178 22 in in ADP ap-1394 178 23 non non ADJ ap-1394 178 24 - - ADJ ap-1394 178 25 hermitian hermitian ADJ ap-1394 178 26 hamiltonians hamiltonian NOUN ap-1394 178 27 having have VERB ap-1394 178 28 pt pt PROPN ap-1394 178 29 symmetry symmetry NOUN ap-1394 178 30 , , PUNCT ap-1394 178 31 phys phy NOUN ap-1394 178 32 . . PUNCT ap-1394 179 1 rev rev PROPN ap-1394 179 2 . . PROPN ap-1394 179 3 lett lett PROPN ap-1394 179 4 . . PROPN ap-1394 180 1 80 80 NUM ap-1394 180 2 , , PUNCT ap-1394 180 3 1998 1998 NUM ap-1394 180 4 , , PUNCT ap-1394 180 5 5 5 NUM ap-1394 180 6 243 243 NUM ap-1394 180 7 . . PUNCT ap-1394 181 1 [ [ X ap-1394 181 2 8 8 NUM ap-1394 181 3 ] ] PUNCT ap-1394 181 4 lévai lévai NOUN ap-1394 181 5 , , PUNCT ap-1394 181 6 g. g. PROPN ap-1394 181 7 , , PUNCT ap-1394 181 8 znojil znojil NOUN ap-1394 181 9 , , PUNCT ap-1394 181 10 m. m. NOUN ap-1394 181 11 : : PUNCT ap-1394 181 12 systematic systematic ADJ ap-1394 181 13 search search NOUN ap-1394 181 14 for for ADP ap-1394 181 15 ptsymmetric ptsymmetric ADJ ap-1394 181 16 potentials potential NOUN ap-1394 181 17 with with ADP ap-1394 181 18 real real ADJ ap-1394 181 19 energy energy NOUN ap-1394 181 20 spectra spectra NOUN ap-1394 181 21 . . PUNCT ap-1394 182 1 j. j. PROPN ap-1394 182 2 phys phys PROPN ap-1394 182 3 . . PUNCT ap-1394 183 1 a a DET ap-1394 183 2 33 33 NUM ap-1394 183 3 , , PUNCT ap-1394 183 4 2000 2000 NUM ap-1394 183 5 , , PUNCT ap-1394 183 6 7 7 NUM ap-1394 183 7 165 165 NUM ap-1394 183 8 . . PUNCT ap-1394 184 1 [ [ X ap-1394 184 2 9 9 NUM ap-1394 184 3 ] ] X ap-1394 184 4 bender bender NOUN ap-1394 184 5 , , PUNCT ap-1394 184 6 c. c. PROPN ap-1394 184 7 m. m. PROPN ap-1394 184 8 , , PUNCT ap-1394 184 9 brody brody PROPN ap-1394 184 10 , , PUNCT ap-1394 184 11 d. d. PROPN ap-1394 184 12 c. c. PROPN ap-1394 184 13 , , PUNCT ap-1394 184 14 jones jones PROPN ap-1394 184 15 , , PUNCT ap-1394 184 16 h. h. PROPN ap-1394 184 17 f. f. PROPN ap-1394 184 18 : : PUNCT ap-1394 184 19 complex complex ADJ ap-1394 184 20 extension extension NOUN ap-1394 184 21 of of ADP ap-1394 184 22 quantum quantum ADJ ap-1394 184 23 mechanics mechanic NOUN ap-1394 184 24 , , PUNCT ap-1394 184 25 phys phy NOUN ap-1394 184 26 . . PUNCT ap-1394 185 1 rev rev PROPN ap-1394 185 2 . . PROPN ap-1394 185 3 lett lett PROPN ap-1394 185 4 . . PROPN ap-1394 185 5 89 89 NUM ap-1394 185 6 , , PUNCT ap-1394 185 7 2002 2002 NUM ap-1394 185 8 , , PUNCT ap-1394 185 9 270 270 NUM ap-1394 185 10 401 401 NUM ap-1394 185 11 . . PUNCT ap-1394 186 1 [ [ X ap-1394 186 2 10 10 NUM ap-1394 186 3 ] ] X ap-1394 186 4 mostafazadeh mostafazadeh NOUN ap-1394 186 5 , , PUNCT ap-1394 186 6 a. a. NOUN ap-1394 186 7 : : PUNCT ap-1394 186 8 pseudo pseudo NOUN ap-1394 186 9 - - NOUN ap-1394 186 10 hermiticity hermiticity NOUN ap-1394 186 11 versus versus ADP ap-1394 186 12 pt pt NOUN ap-1394 186 13 - - PUNCT ap-1394 186 14 symmetry symmetry NOUN ap-1394 186 15 iii iii NOUN ap-1394 186 16 : : PUNCT ap-1394 186 17 equivalence equivalence NOUN ap-1394 186 18 of of ADP ap-1394 186 19 pseudohermiticity pseudohermiticity NOUN ap-1394 186 20 and and CCONJ ap-1394 186 21 the the DET ap-1394 186 22 presence presence NOUN ap-1394 186 23 of of ADP ap-1394 186 24 antilinear antilinear ADJ ap-1394 186 25 symmetries symmetry NOUN ap-1394 186 26 , , PUNCT ap-1394 186 27 j. j. PROPN ap-1394 186 28 math math PROPN ap-1394 186 29 . . PUNCT ap-1394 187 1 phys phy NOUN ap-1394 187 2 . . PUNCT ap-1394 188 1 43 43 NUM ap-1394 188 2 , , PUNCT ap-1394 188 3 2002 2002 NUM ap-1394 188 4 , , PUNCT ap-1394 188 5 3 3 NUM ap-1394 188 6 944 944 NUM ap-1394 188 7 . . PUNCT ap-1394 189 1 [ [ X ap-1394 189 2 11 11 NUM ap-1394 189 3 ] ] PUNCT ap-1394 189 4 oko oko PROPN ap-1394 189 5 lowicz lowicz NOUN ap-1394 189 6 , , PUNCT ap-1394 189 7 j. j. PROPN ap-1394 189 8 , , PUNCT ap-1394 189 9 p p NOUN ap-1394 189 10 loszajczak loszajczak NOUN ap-1394 189 11 , , PUNCT ap-1394 189 12 m. m. NOUN ap-1394 189 13 , , PUNCT ap-1394 189 14 rotter rotter NOUN ap-1394 189 15 , , PUNCT ap-1394 189 16 i. i. NOUN ap-1394 189 17 : : PUNCT ap-1394 189 18 dynamics dynamic NOUN ap-1394 189 19 of of ADP ap-1394 189 20 quantum quantum NOUN ap-1394 189 21 systems system NOUN ap-1394 189 22 embedded embed VERB ap-1394 189 23 in in ADP ap-1394 189 24 a a DET ap-1394 189 25 continuum continuum ADJ ap-1394 189 26 , , PUNCT ap-1394 189 27 phys phy NOUN ap-1394 189 28 . . PUNCT ap-1394 190 1 rep rep PROPN ap-1394 190 2 . . PROPN ap-1394 190 3 374 374 NUM ap-1394 190 4 , , PUNCT ap-1394 190 5 2003 2003 NUM ap-1394 190 6 , , PUNCT ap-1394 190 7 271 271 NUM ap-1394 190 8 . . PUNCT ap-1394 191 1 [ [ X ap-1394 191 2 12 12 NUM ap-1394 191 3 ] ] PUNCT ap-1394 191 4 jones jones PROPN ap-1394 191 5 , , PUNCT ap-1394 191 6 h. h. PROPN ap-1394 191 7 f. f. PROPN ap-1394 191 8 , , PUNCT ap-1394 191 9 rivers river NOUN ap-1394 191 10 , , PUNCT ap-1394 191 11 r. r. PROPN ap-1394 191 12 j. j. PROPN ap-1394 191 13 : : PUNCT ap-1394 191 14 which which DET ap-1394 191 15 green green ADJ ap-1394 191 16 functions function NOUN ap-1394 191 17 does do AUX ap-1394 191 18 the the DET ap-1394 191 19 path path NOUN ap-1394 191 20 integral integral ADJ ap-1394 191 21 for for SCONJ ap-1394 191 22 quasi quasi ADJ ap-1394 191 23 - - ADJ ap-1394 191 24 hermitian hermitian ADJ ap-1394 191 25 hamiltonians hamiltonian NOUN ap-1394 191 26 represent represent VERB ap-1394 191 27 ? ? PUNCT ap-1394 192 1 phys phy NOUN ap-1394 192 2 . . PUNCT ap-1394 193 1 lett lett PROPN ap-1394 193 2 . . PUNCT ap-1394 194 1 a a DET ap-1394 194 2 373 373 NUM ap-1394 194 3 , , PUNCT ap-1394 194 4 2009 2009 NUM ap-1394 194 5 , , PUNCT ap-1394 194 6 3 3 NUM ap-1394 194 7 304 304 NUM ap-1394 194 8 . . PUNCT ap-1394 195 1 [ [ X ap-1394 195 2 13 13 NUM ap-1394 195 3 ] ] X ap-1394 195 4 dorey dorey PROPN ap-1394 195 5 , , PUNCT ap-1394 195 6 p. p. NOUN ap-1394 195 7 , , PUNCT ap-1394 195 8 dunning dunning NOUN ap-1394 195 9 , , PUNCT ap-1394 195 10 c. c. NOUN ap-1394 195 11 , , PUNCT ap-1394 195 12 lishman lishman NOUN ap-1394 195 13 , , PUNCT ap-1394 195 14 a. a. NOUN ap-1394 195 15 , , PUNCT ap-1394 195 16 tateo tateo NOUN ap-1394 195 17 , , PUNCT ap-1394 195 18 r. r. PROPN ap-1394 195 19 : : PUNCT ap-1394 195 20 pt pt PROPN ap-1394 195 21 symmetry symmetry NOUN ap-1394 195 22 breaking break VERB ap-1394 195 23 and and CCONJ ap-1394 195 24 exceptional exceptional ADJ ap-1394 195 25 points point NOUN ap-1394 195 26 for for ADP ap-1394 195 27 a a DET ap-1394 195 28 class class NOUN ap-1394 195 29 of of ADP ap-1394 195 30 inhomogeneous inhomogeneous ADJ ap-1394 195 31 complex complex ADJ ap-1394 195 32 potentials potential NOUN ap-1394 195 33 , , PUNCT ap-1394 195 34 j. j. PROPN ap-1394 195 35 phys phys PROPN ap-1394 195 36 . . PUNCT ap-1394 196 1 a a DET ap-1394 196 2 42 42 NUM ap-1394 196 3 , , PUNCT ap-1394 196 4 2009 2009 NUM ap-1394 196 5 , , PUNCT ap-1394 196 6 465 465 NUM ap-1394 196 7 302 302 NUM ap-1394 196 8 . . PUNCT ap-1394 197 1 [ [ X ap-1394 197 2 14 14 NUM ap-1394 197 3 ] ] PUNCT ap-1394 197 4 witten witten PROPN ap-1394 197 5 , , PUNCT ap-1394 197 6 e. e. PROPN ap-1394 197 7 : : PUNCT ap-1394 197 8 a a DET ap-1394 197 9 new new ADJ ap-1394 197 10 look look NOUN ap-1394 197 11 at at ADP ap-1394 197 12 the the DET ap-1394 197 13 path path NOUN ap-1394 197 14 integral integral ADJ ap-1394 197 15 of of ADP ap-1394 197 16 quantum quantum NOUN ap-1394 197 17 mechanics mechanic NOUN ap-1394 197 18 . . PUNCT ap-1394 198 1 in in ADP ap-1394 198 2 surveys survey NOUN ap-1394 198 3 in in ADP ap-1394 198 4 differential differential ADJ ap-1394 198 5 geometry geometry NOUN ap-1394 198 6 xv xv PROPN ap-1394 198 7 . . PUNCT ap-1394 198 8 mrowka mrowka NOUN ap-1394 198 9 , , PUNCT ap-1394 198 10 t. t. PROPN ap-1394 198 11 , , PUNCT ap-1394 198 12 yau yau PROPN ap-1394 198 13 , , PUNCT ap-1394 198 14 s.-t s.-t NOUN ap-1394 198 15 . . PUNCT ap-1394 199 1 ( ( PUNCT ap-1394 199 2 eds ed NOUN ap-1394 199 3 . . PUNCT ap-1394 199 4 ) ) PUNCT ap-1394 199 5 boston boston PROPN ap-1394 199 6 : : PUNCT ap-1394 199 7 international international ADJ ap-1394 199 8 press press NOUN ap-1394 199 9 , , PUNCT ap-1394 199 10 2011 2011 NUM ap-1394 199 11 . . PUNCT ap-1394 200 1 [ [ X ap-1394 200 2 15 15 NUM ap-1394 200 3 ] ] X ap-1394 200 4 graefe graefe NOUN ap-1394 200 5 , , PUNCT ap-1394 200 6 e. e. PROPN ap-1394 200 7 m. m. PROPN ap-1394 200 8 , , PUNCT ap-1394 200 9 korsch korsch PROPN ap-1394 200 10 , , PUNCT ap-1394 200 11 h. h. PROPN ap-1394 200 12 j. j. PROPN ap-1394 200 13 , , PUNCT ap-1394 200 14 niederle niederle PROPN ap-1394 200 15 , , PUNCT ap-1394 200 16 a. a. PROPN ap-1394 200 17 e. e. PROPN ap-1394 200 18 : : PUNCT ap-1394 200 19 quantum quantum ADJ ap-1394 200 20 - - ADJ ap-1394 200 21 classical classical ADJ ap-1394 200 22 correspondence correspondence NOUN ap-1394 200 23 for for ADP ap-1394 200 24 a a DET ap-1394 200 25 nonhermitian nonhermitian ADJ ap-1394 200 26 bose bose PROPN ap-1394 200 27 - - PUNCT ap-1394 200 28 hubbard hubbard NOUN ap-1394 200 29 dimer dimer NOUN ap-1394 200 30 , , PUNCT ap-1394 200 31 phys phy NOUN ap-1394 200 32 . . PUNCT ap-1394 201 1 rev rev PROPN ap-1394 201 2 . . PUNCT ap-1394 202 1 a a DET ap-1394 202 2 82 82 NUM ap-1394 202 3 , , PUNCT ap-1394 202 4 2010 2010 NUM ap-1394 202 5 , , PUNCT ap-1394 202 6 013 013 NUM ap-1394 202 7 629 629 NUM ap-1394 202 8 . . PUNCT ap-1394 203 1 [ [ X ap-1394 203 2 16 16 NUM ap-1394 203 3 ] ] X ap-1394 203 4 günther günther PROPN ap-1394 203 5 , , PUNCT ap-1394 203 6 u. u. PROPN ap-1394 203 7 , , PUNCT ap-1394 203 8 kuzhel kuzhel PROPN ap-1394 203 9 , , PUNCT ap-1394 203 10 s. s. PROPN ap-1394 203 11 : : PUNCT ap-1394 203 12 pt pt PROPN ap-1394 203 13 - - PUNCT ap-1394 203 14 symmetry symmetry NOUN ap-1394 203 15 , , PUNCT ap-1394 203 16 cartan cartan ADJ ap-1394 203 17 decompositions decomposition NOUN ap-1394 203 18 , , PUNCT ap-1394 203 19 lie lie VERB ap-1394 203 20 triple triple ADJ ap-1394 203 21 systems system NOUN ap-1394 203 22 and and CCONJ ap-1394 203 23 krein krein ADJ ap-1394 203 24 space space NOUN ap-1394 203 25 - - PUNCT ap-1394 203 26 related relate VERB ap-1394 203 27 clifford clifford PROPN ap-1394 203 28 algebras algebras PROPN ap-1394 203 29 . . PUNCT ap-1394 204 1 j. j. PROPN ap-1394 204 2 phys phys PROPN ap-1394 204 3 . . PUNCT ap-1394 205 1 a a DET ap-1394 205 2 43 43 NUM ap-1394 205 3 , , PUNCT ap-1394 205 4 2010 2010 NUM ap-1394 205 5 , , PUNCT ap-1394 205 6 39 39 NUM ap-1394 205 7 002 002 NUM ap-1394 205 8 . . PUNCT ap-1394 206 1 [ [ X ap-1394 206 2 17 17 NUM ap-1394 206 3 ] ] X ap-1394 206 4 moiseyev moiseyev NOUN ap-1394 206 5 , , PUNCT ap-1394 206 6 n. n. NOUN ap-1394 206 7 : : PUNCT ap-1394 206 8 non non ADJ ap-1394 206 9 - - ADJ ap-1394 206 10 hermitian hermitian ADJ ap-1394 206 11 quantum quantum ADJ ap-1394 206 12 mechanics mechanic NOUN ap-1394 206 13 , , PUNCT ap-1394 206 14 cambridge cambridge NOUN ap-1394 206 15 : : PUNCT ap-1394 206 16 cambridge cambridge PROPN ap-1394 206 17 university university PROPN ap-1394 206 18 press press NOUN ap-1394 206 19 , , PUNCT ap-1394 206 20 2011 2011 NUM ap-1394 206 21 . . PUNCT ap-1394 207 1 [ [ X ap-1394 207 2 18 18 NUM ap-1394 207 3 ] ] SYM ap-1394 207 4 nesterov nesterov ADJ ap-1394 207 5 , , PUNCT ap-1394 207 6 a. a. NOUN ap-1394 207 7 i. i. PROPN ap-1394 207 8 : : PUNCT ap-1394 207 9 non non ADJ ap-1394 207 10 - - ADJ ap-1394 207 11 hermitian hermitian ADJ ap-1394 207 12 quantum quantum NOUN ap-1394 207 13 systems system NOUN ap-1394 207 14 and and CCONJ ap-1394 207 15 time time NOUN ap-1394 207 16 - - PUNCT ap-1394 207 17 optimal optimal ADJ ap-1394 207 18 quantum quantum ADJ ap-1394 207 19 evolution evolution NOUN ap-1394 207 20 . . PUNCT ap-1394 208 1 sigma sigma PROPN ap-1394 208 2 , , PUNCT ap-1394 208 3 5 5 NUM ap-1394 208 4 , , PUNCT ap-1394 208 5 2009 2009 NUM ap-1394 208 6 , , PUNCT ap-1394 208 7 069 069 NUM ap-1394 208 8 . . PUNCT ap-1394 209 1 [ [ X ap-1394 209 2 19 19 NUM ap-1394 209 3 ] ] X ap-1394 209 4 brody brody PROPN ap-1394 209 5 , , PUNCT ap-1394 209 6 d. d. PROPN ap-1394 209 7 c. c. PROPN ap-1394 209 8 , , PUNCT ap-1394 209 9 graefe graefe PROPN ap-1394 209 10 , , PUNCT ap-1394 209 11 e. e. PROPN ap-1394 209 12 m. m. PROPN ap-1394 209 13 : : PUNCT ap-1394 209 14 on on ADP ap-1394 209 15 complexified complexifie VERB ap-1394 209 16 mechanics mechanic NOUN ap-1394 209 17 and and CCONJ ap-1394 209 18 coquaternions coquaternion NOUN ap-1394 209 19 , , PUNCT ap-1394 209 20 j. j. PROPN ap-1394 209 21 phys phys PROPN ap-1394 209 22 . . PUNCT ap-1394 210 1 a a DET ap-1394 210 2 44 44 NUM ap-1394 210 3 , , PUNCT ap-1394 210 4 2011 2011 NUM ap-1394 210 5 , , PUNCT ap-1394 210 6 072 072 NUM ap-1394 210 7 001 001 NUM ap-1394 210 8 . . PUNCT ap-1394 211 1 [ [ X ap-1394 211 2 20 20 NUM ap-1394 211 3 ] ] SYM ap-1394 211 4 finkelstein finkelstein PROPN ap-1394 211 5 , , PUNCT ap-1394 211 6 d. d. PROPN ap-1394 211 7 , , PUNCT ap-1394 211 8 jaueh jaueh PROPN ap-1394 211 9 , , PUNCT ap-1394 211 10 j. j. PROPN ap-1394 211 11 m. m. PROPN ap-1394 211 12 , , PUNCT ap-1394 211 13 schiminovieh schiminovieh PROPN ap-1394 211 14 , , PUNCT ap-1394 211 15 s. s. PROPN ap-1394 211 16 , , PUNCT ap-1394 211 17 speiser speiser PROPN ap-1394 211 18 , , PUNCT ap-1394 211 19 d. d. PROPN ap-1394 211 20 : : PUNCT ap-1394 211 21 foundations foundation NOUN ap-1394 211 22 of of ADP ap-1394 211 23 quaternion quaternion ADJ ap-1394 211 24 quantum quantum ADJ ap-1394 211 25 mechanics mechanic NOUN ap-1394 211 26 , , PUNCT ap-1394 211 27 j. j. PROPN ap-1394 211 28 math math PROPN ap-1394 211 29 . . PUNCT ap-1394 212 1 phys phy NOUN ap-1394 212 2 . . PUNCT ap-1394 213 1 3 3 NUM ap-1394 213 2 , , PUNCT ap-1394 213 3 1962 1962 NUM ap-1394 213 4 , , PUNCT ap-1394 213 5 207 207 NUM ap-1394 213 6 . . PUNCT ap-1394 214 1 [ [ X ap-1394 214 2 21 21 NUM ap-1394 214 3 ] ] X ap-1394 214 4 adler adler PROPN ap-1394 214 5 , , PUNCT ap-1394 214 6 s. s. PROPN ap-1394 214 7 l. l. PROPN ap-1394 214 8 : : PUNCT ap-1394 214 9 quaternionic quaternionic ADJ ap-1394 214 10 quantum quantum ADJ ap-1394 214 11 mechanics mechanic NOUN ap-1394 214 12 and and CCONJ ap-1394 214 13 quantum quantum NOUN ap-1394 214 14 fields field NOUN ap-1394 214 15 . . PUNCT ap-1394 215 1 oxford oxford NOUN ap-1394 215 2 : : PUNCT ap-1394 215 3 oxford oxford PROPN ap-1394 215 4 university university PROPN ap-1394 215 5 press press NOUN ap-1394 215 6 , , PUNCT ap-1394 215 7 1995 1995 NUM ap-1394 215 8 . . PUNCT ap-1394 216 1 [ [ X ap-1394 216 2 22 22 NUM ap-1394 216 3 ] ] X ap-1394 216 4 brody brody PROPN ap-1394 216 5 , , PUNCT ap-1394 216 6 d. d. PROPN ap-1394 216 7 c. c. PROPN ap-1394 216 8 , , PUNCT ap-1394 216 9 graefe graefe PROPN ap-1394 216 10 , , PUNCT ap-1394 216 11 e. e. PROPN ap-1394 216 12 m. m. PROPN ap-1394 216 13 : : PUNCT ap-1394 216 14 six six NUM ap-1394 216 15 - - PUNCT ap-1394 216 16 dimensional dimensional ADJ ap-1394 216 17 space space NOUN ap-1394 216 18 - - PUNCT ap-1394 216 19 time time NOUN ap-1394 216 20 from from ADP ap-1394 216 21 quaternionic quaternionic ADJ ap-1394 216 22 quantum quantum NOUN ap-1394 216 23 mechanics mechanic NOUN ap-1394 216 24 . . PUNCT ap-1394 216 25 2011 2011 NUM ap-1394 216 26 , , PUNCT ap-1394 216 27 arxiv:1105.3604 arxiv:1105.3604 VERB ap-1394 216 28 . . PUNCT ap-1394 217 1 [ [ X ap-1394 217 2 23 23 NUM ap-1394 217 3 ] ] X ap-1394 217 4 kisil kisil NOUN ap-1394 217 5 , , PUNCT ap-1394 217 6 v. v. PROPN ap-1394 218 1 v. v. ADJ ap-1394 218 2 : : PUNCT ap-1394 218 3 erlangen erlangen PROPN ap-1394 218 4 programme programme PROPN ap-1394 218 5 at at ADP ap-1394 218 6 large large ADJ ap-1394 218 7 3.1 3.1 NUM ap-1394 218 8 : : PUNCT ap-1394 218 9 hypercomplex hypercomplex ADJ ap-1394 218 10 representations representation NOUN ap-1394 218 11 of of ADP ap-1394 218 12 the the DET ap-1394 218 13 heisenberg heisenberg PROPN ap-1394 218 14 group group NOUN ap-1394 218 15 and and CCONJ ap-1394 218 16 mechanics mechanic NOUN ap-1394 218 17 . . PUNCT ap-1394 219 1 2010 2010 NUM ap-1394 219 2 , , PUNCT ap-1394 219 3 arxiv:1005.5057v2 arxiv:1005.5057v2 NOUN ap-1394 219 4 [ [ X ap-1394 219 5 24 24 NUM ap-1394 219 6 ] ] X ap-1394 219 7 cockle cockle PROPN ap-1394 219 8 , , PUNCT ap-1394 219 9 j. j. PROPN ap-1394 219 10 : : PUNCT ap-1394 219 11 on on ADP ap-1394 219 12 systems system NOUN ap-1394 219 13 of of ADP ap-1394 219 14 algebra algebra NOUN ap-1394 219 15 involving involve VERB ap-1394 219 16 more more ADJ ap-1394 219 17 than than ADP ap-1394 219 18 one one NUM ap-1394 219 19 imaginary imaginary ADJ ap-1394 219 20 ; ; PUNCT ap-1394 219 21 and and CCONJ ap-1394 219 22 on on ADP ap-1394 219 23 equations equation NOUN ap-1394 219 24 of of ADP ap-1394 219 25 the the DET ap-1394 219 26 fifth fifth ADJ ap-1394 219 27 degree degree NOUN ap-1394 219 28 , , PUNCT ap-1394 220 1 phil phil PROPN ap-1394 220 2 . . PROPN ap-1394 220 3 magazine magazine PROPN ap-1394 220 4 35 35 NUM ap-1394 220 5 , , PUNCT ap-1394 220 6 1849 1849 NUM ap-1394 220 7 , , PUNCT ap-1394 220 8 434 434 NUM ap-1394 220 9 . . PUNCT ap-1394 221 1 [ [ X ap-1394 221 2 25 25 NUM ap-1394 221 3 ] ] X ap-1394 221 4 wang wang PROPN ap-1394 221 5 , , PUNCT ap-1394 221 6 q. q. PROPN ap-1394 221 7 , , PUNCT ap-1394 221 8 chia chia PROPN ap-1394 221 9 , , PUNCT ap-1394 221 10 s. s. PROPN ap-1394 221 11 , , PUNCT ap-1394 221 12 zhang zhang PROPN ap-1394 221 13 , , PUNCT ap-1394 221 14 j. j. PROPN ap-1394 221 15 : : PUNCT ap-1394 221 16 pt pt PROPN ap-1394 221 17 symmetry symmetry NOUN ap-1394 221 18 as as ADP ap-1394 221 19 a a DET ap-1394 221 20 generalisation generalisation NOUN ap-1394 221 21 of of ADP ap-1394 221 22 hermiticity hermiticity PROPN ap-1394 221 23 . . PUNCT ap-1394 222 1 j. j. PROPN ap-1394 222 2 phys phys PROPN ap-1394 222 3 . . PUNCT ap-1394 223 1 a a DET ap-1394 223 2 43 43 NUM ap-1394 223 3 , , PUNCT ap-1394 223 4 2010 2010 NUM ap-1394 223 5 , , PUNCT ap-1394 223 6 295 295 NUM ap-1394 223 7 301 301 NUM ap-1394 223 8 . . PUNCT ap-1394 224 1 [ [ X ap-1394 224 2 26 26 NUM ap-1394 224 3 ] ] X ap-1394 224 4 clifford clifford PROPN ap-1394 224 5 , , PUNCT ap-1394 224 6 w. w. PROPN ap-1394 224 7 k. k. PROPN ap-1394 224 8 : : PUNCT ap-1394 224 9 applications application NOUN ap-1394 224 10 of of ADP ap-1394 224 11 grassmann grassmann PROPN ap-1394 224 12 ’s ’s PART ap-1394 224 13 extensive extensive ADJ ap-1394 224 14 algebra algebra NOUN ap-1394 224 15 . . PUNCT ap-1394 225 1 am be AUX ap-1394 225 2 . . PUNCT ap-1394 226 1 j. j. PROPN ap-1394 226 2 math math PROPN ap-1394 226 3 . . PUNCT ap-1394 227 1 1 1 NUM ap-1394 227 2 , , PUNCT ap-1394 227 3 1878 1878 NUM ap-1394 227 4 , , PUNCT ap-1394 227 5 350 350 NUM ap-1394 227 6 . . PUNCT ap-1394 228 1 [ [ X ap-1394 228 2 27 27 NUM ap-1394 228 3 ] ] X ap-1394 228 4 hudson hudson PROPN ap-1394 228 5 , , PUNCT ap-1394 228 6 r. r. PROPN ap-1394 228 7 l. l. PROPN ap-1394 228 8 : : PUNCT ap-1394 228 9 generalised generalise VERB ap-1394 228 10 translation translation NOUN ap-1394 228 11 - - PUNCT ap-1394 228 12 invariant invariant ADJ ap-1394 228 13 mechanics mechanic NOUN ap-1394 228 14 . . PUNCT ap-1394 229 1 d. d. PROPN ap-1394 229 2 phil phil PROPN ap-1394 229 3 . . PUNCT ap-1394 230 1 thesis thesis NOUN ap-1394 230 2 , , PUNCT ap-1394 230 3 bodleian bodleian ADJ ap-1394 230 4 library library NOUN ap-1394 230 5 , , PUNCT ap-1394 230 6 oxford oxford PROPN ap-1394 230 7 , , PUNCT ap-1394 230 8 1966 1966 NUM ap-1394 230 9 . . PUNCT ap-1394 231 1 19 19 NUM ap-1394 231 2 acta acta PROPN ap-1394 231 3 polytechnica polytechnica PROPN ap-1394 231 4 vol vol NOUN ap-1394 231 5 . . PUNCT ap-1394 231 6 51 51 NUM ap-1394 231 7 no no INTJ ap-1394 231 8 . . PUNCT ap-1394 232 1 4/2011 4/2011 NUM ap-1394 233 1 [ [ X ap-1394 233 2 28 28 NUM ap-1394 233 3 ] ] X ap-1394 233 4 kocik kocik PROPN ap-1394 233 5 , , PUNCT ap-1394 233 6 j. j. PROPN ap-1394 233 7 : : PUNCT ap-1394 233 8 duplex duplex NOUN ap-1394 233 9 numbers number NOUN ap-1394 233 10 , , PUNCT ap-1394 233 11 diffusion diffusion NOUN ap-1394 233 12 systems system NOUN ap-1394 233 13 , , PUNCT ap-1394 233 14 and and CCONJ ap-1394 233 15 generalized generalized ADJ ap-1394 233 16 quantum quantum ADJ ap-1394 233 17 mechanics mechanic NOUN ap-1394 233 18 , , PUNCT ap-1394 233 19 int int NOUN ap-1394 233 20 . . PUNCT ap-1394 234 1 j. j. PROPN ap-1394 234 2 theo theo PROPN ap-1394 234 3 . . PUNCT ap-1394 235 1 phys phy NOUN ap-1394 235 2 . . PUNCT ap-1394 236 1 38 38 NUM ap-1394 236 2 , , PUNCT ap-1394 236 3 1999 1999 NUM ap-1394 236 4 , , PUNCT ap-1394 236 5 2221 2221 NUM ap-1394 236 6 . . PUNCT ap-1394 237 1 dorje dorje PROPN ap-1394 237 2 c. c. PROPN ap-1394 237 3 brody brody PROPN ap-1394 237 4 mathematical mathematical PROPN ap-1394 237 5 sciences sciences PROPN ap-1394 237 6 brunel brunel PROPN ap-1394 237 7 university university PROPN ap-1394 237 8 uxbridge uxbridge VERB ap-1394 237 9 ub8 ub8 PROPN ap-1394 237 10 3ph 3ph NOUN ap-1394 237 11 , , PUNCT ap-1394 237 12 uk uk PROPN ap-1394 237 13 eva eva PROPN ap-1394 237 14 - - PUNCT ap-1394 237 15 maria maria PROPN ap-1394 237 16 graefe graefe PROPN ap-1394 237 17 department department PROPN ap-1394 237 18 of of ADP ap-1394 237 19 mathematics mathematics PROPN ap-1394 237 20 imperial imperial PROPN ap-1394 237 21 college college PROPN ap-1394 237 22 london london PROPN ap-1394 237 23 london london PROPN ap-1394 237 24 sw7 sw7 PROPN ap-1394 237 25 2az 2az PROPN ap-1394 237 26 , , PUNCT ap-1394 237 27 uk uk PROPN ap-1394 237 28 20 20 NUM