ap-3-10.dvi


Acta Polytechnica Vol. 50 No. 3/2010

On Global and Nonlinear Symmetries in Quantum Mechanics

H.-D. Doebner

1 Prolog

I firstmet Jiri Niederle around 40 years ago in the In-
ternational Centre for Theoretical Physics at Trieste.
During this time the Centre was located in a modern
building at Piazza Oberdan, the Director was Abdus
Salam and his deputy was Paolo Budini. Jiri and I
were fellows in a group under the indirect guidance of
AsimBarut andChris Fronsdal. I remember some our
colleagues in this group: ArnoBohm,RichardRaczka,
Moshe Flato. It was the period in which Abdus asked
the scientist in the centre ‘topush forwardthe frontiers
of knowledge’, especially in particle physics through
models based on group theory; Ũ(12) was fashionable
in these times.
Some of us expected a description of the geomet-

rical structure behind fundamental quantum physics
through groups, their Lie algebras and their represen-
tations to be too narrow and not flexible enough to
model real physical classical and quantum systems:
If the group and their linear representation are cho-
sen, there isnot enough freedomtoaccommodate their
characteristic properties. Hence, wewere interested to
apply a framework which is beyond groups. We tried
e.g. nonlinear and nonintegrable Lie-algebra represen-
tations, nonseparable Hilbert spaces and Lie algebras
over unusual number fields; we studied differential ge-
ometric methods which reflect local as well as global
geometrical, algebraic, analytic and symmetry prop-
erties which are more general compared with group
theory.
Jiri Niederle and JoukoMickelssonwere interested

in the construction of nonlinear representations,which
opens the plethora of unknown possibilities to model
Lie symmetries of quantum systems. These represen-
tations contain nonlinear generators acting in a linear
representation space. One has to specify the type of
‘nonlinearity’ and their physical interpretation.
Jiri Tolar and Imyself intended tomodel quantum

systems localizedandmovingona smoothmanifold M
with (classical) position andmomentum observable O
which span their kinematics K(M) through a quanti-
zation Q(M, K) of K(M) –BORELkinematics – and,
after the introductionof a time t dependence, their dy-
namical properties – BOREL quantization – through
e.g. as quantum analogous to the classical case.
It is reasonable to connect a paper on the occasion

of the 70th birthday of Jiri Niederle with a retrospec-

tive view on both of our attempts. We thought in Tri-
este that these ideas – the utilization of nonlinear rep-
resentations and the application of global methods –
were not directly related to each other. However, it
turns out now that this is not the case.
A discussion of these relations is the topic of my

paper: The global structure of the set {Q(M, K)} of
quantizations of the kinematics in aHilbert space H is
connected with a nonlinear representation of a matrix
group G which leads also to a quantization of other
observables through nonlinear operators. One obtains
e.g. for M = R3 a nonlinear representation of the cen-
tral extensionof the inhomogeneousGalilei groupG(3)
and furthermore nonlinear Schrödinger equationswith
given potentials which were also derived in another
context in [1, 2].

2 Preview
Weconsider scalarnonrelativistic systems localized

andmoving ona smoothmanifoldwithout internal de-
grees of freedom and external fields (2-forms on M)
and their quantizations (quantummaps) on a suitable
Hilbert space H:

The set of unitary inequivalent quantummaps Q
of systems on M with kinmatic K(M)

{Qα,D(M, K)}

is labelled through two quantum numbers α and D
which depend on topological and global properties of
M and K(M) [3, 4], for reviews see [5, 6]. We explain
this in section 3.

In the case M = R3 (with trivial α and
a real number D), presented in section 4, the set
{QD(R3, K)} is shown to carry a physicallymotivated
nonlinear representation NG of a 2×2 matrix group
G which act on a domain D ∈ H

NG ∈ N:ψ −→ N[ψ]. (1)

This implies that unitary inequivalentquantizations in
QD(O), O = f, X, for different D are related through
tangent maps with N (see also [4])

d
d�
(N[ψ + i�QD(O)ψ])�=0 = i Q

D′(O) · N[ψ]. (2)

This construction leads also to nonlinear operators
from elements of order ≥ 2 in esa polynomials in Q0
and P0.

A Lecture on the Occasion of the 70th Birthday of Jiri Niederle

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Acta Polytechnica Vol. 50 No. 3/2010

3 Borel kinematic K(M) and
its quantization

3.1 K(M) as global kinematical
algebra

Our systems are localized and moving on a smooth
manifold M.
To model their localizations we consider a set of

regions on M such that this set should contain infor-
mation on the probabilities to observe the system in a
given region. A sufficiently large canonical set of re-
gions is the σ-algebra B(M) of Borel sets in M. We
choose B(M) as set of position observables.
Smoothmotions of systems localized in B ∈ B(M)

can be described canonically through flows φ on M

B �→ B′ = φXs (B)

parameterized with s and characterized through in-
finitesimal generators X which are contained in the
set V ect0(M) of smooth complete vectorfields in M.
These covers a large class of motions; their generators
X can be chosen as momentum observables. Collect-
ing the results we have:
• The kinematical situation of the system is de-

scribed with a flow model in M. Position observables
are modelled with the σ-algebra B(M) and momen-
tum observableswith vectorfields V ect0(M). The col-
lection of these observable

K(M)= (B(M), V ect0(M)) (3)

contains through the flow model all possible positions
andmomenta as global properties of themovingBorel
field B(M). This justifies the name ‘Borel’ kinematic.
This notion can be generalized [4, 7, 8] to systems

with k internal degrees of freedom and external fields
(2-forms on M).
The kinematics K(M) is a time independent

quantity. If a time dependence t is introduced to
parametrize the motion of the system, e.g. through
t dependent states, the quantization Q of K(M) leads
to evolution equations which select, with e.g. initial
values, the t dependence of the matrix element of the
quantized kinematical observable (see e.g. [1, 9, 10]).

3.2 Quantizations of K(M)

We use ‘quantization’ in the following sense: Con-
sider a set of classical objects, e.g. K(M), a separable
Hilbert space H and the set SA(H) of esa operators
(and hence linear) in H. The states of the system are
modelled through H. We define ‘quantization’ as a
map Q (quantummap) of this classical object on a cdi
domain in SA(H) with certain properties depending

on the structure of the system. Hence a quantization
of K(M) is a quantum map Q(M, K)= (Q,P)

Q:K(M)� (B, X) �→ (Q(B),P(X) ∈ SA(H) (4)

on a cdi domain D in H.

3.2.1 Quantizations of Borel fields B(M) and
Algebraic Properties of K(M)

Following the geometrical background of the flow
model and the above notion we interpret the map

Q:B �→ Q(B) ∈ SA(H), B ∈ B(M) (5)

through thematrix elements ofQ(B) in a (pure) state
with ψ ∈ D

μϕ(B)= (ψ,Q(B)ψ))/(ψ, ψ) (6)

as the probability to find the system in this state lo-
calized in B ∈ B(M). Therefore we assume μψ:B �−→
μψ(B) to be a probability measure on B(M) which
implies that Q(B) is a positive operator valued (pov)
measure on B(M). We assume that the position ob-
servable B and also their traumatizations Q(B) are
commutative1 and get a projective pov valued mea-
sure Q. If H is realized as L2(M, ν) (ν is a standard
measure on M) the Q(B) act, up to unitary equiva-
lence, as a multiplication operator with the character-
istic function of B

Q(B)ψ = χ(B)ψ. (7)

Finallywe induce aquantummapQof the set C∞(M)
of real smooth functions f(m) on m ∈ M via the spec-
tral theorem

Q:C∞(M) � f �→ Q(f(m)= f(m) ∈ SA(L2(M, ν)).
(8)

The model for position observableswith smooth func-
tions f ∈ C∞(M)) instead with Borel sets B ∈ B(M)
ismotivatedbecause one can equivalentlywrite K(M)
as

K(M)= (C∞(M), V ect0(M)) (9)

which implies an algebraic structure of K(M): the
abelianLie-algebra C∞(M) and the algebra of smooth
vectorfields V ect0(M) which act together on M as
semidirect sum

K(M)= C∞(M)⊕s V ect0(M). (10)

Hence K(M) can be viewed as an (∞ dim) global in-
finite dimensional Lie-algebraic symmetry of a system
on M. It can also be considered as Lie-algebra of an
inhomogeneous subgroup of the diffeomorphism group
Dif f(M) which was used in [11].

1Certain types of noncommutative positions can also be discussed in this approach.

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Acta Polytechnica Vol. 50 No. 3/2010

This symmetry (10) reflects the physics in the flow
model and it is plausible to assume that the symmetry
survives the quantizationmapQand leads after quan-
tization in L2(M, ν) to a partial realization of K(M)
in (10). We add ‘partial’ because two esa elements
in K(M) on D belong to Q(M, K) only if their lin-
ear combinations and commutators are also esa on D.
Collecting the result we have:
• Quantum maps Q of position observable f ∈

C∞(M) lead to Q(f) which act as multiplication op-
erators in L2(M, ν). This allows to view K(M) as
global infinite dimensional Lie-algebraic symmetry of
the system which is assumed to survive the quantum
map.

3.2.2 Quantizations of V ect0(M)

Momentum observables in K(M) are introduced as
flow generators X ∈ V ect0(M) of φX. Following our
above arguments their quantum map P should again
be based on the geometrical roots and must respect
the consistency with Q(f) in (10).

I.

The observable X act in our flowmodel as differential
operators on position observables in C∞(M). Hence
it is plausible to assume that also P(X) ∈ SA(L2(.))
is a finite order differential operator on L2(M, ν).
However, a realization of this assumption is difficult.
This is because a definition of differential operators in
L2(M, ν) needs a notion of the differentiation of com-
plex functions ψ(m) on a smooth manifold M which
are square integrable in respect to themeasure ν. This
measure theoretic property containsno information on
their differentiability. Hence additional assumptions
are necessary to realizemomentum operators P(X) as
PDO’s.
To formulate such assumptions we note that dif-

ferentiation on a smoothmanifold M is given through
its definition, and differentiation on the complex plane
C is a standard notion. This implies technically the
existence of differential structures -D(M) on M and
-D(C) on C. For the differentiation of complex func-
tions on M in L2(M, ν) we need technically a differen-
tial structure -D(M × C) on the point set M × C with
the restrictions

-D(M ×C)/M = -D(M), -D(M ×C)/C = -D(C). (11)

Without going into mathematical details, including
the definition of -D and interesting applications, we
need for P some results on differential structures with
(11). A possibility is to look on complex line bundles
η over M with hermiteanmetric <, >. This is because
there exists with η an isometrically isomorphic com-
plex line bundle η0 with hermitean metric <, >0 and
a differentiable structure -D(M × C) with (11). Hence
we are interested in complex line bundles with metric
<, > with Hermitean connection ∇ and in their clas-

sification up to equivalence. This is well known, we
refer e.g. to [12, 13]. Because we are dealing with a
system without external fields the connection is flat.
The inequivalent line bundles with flat connection are
labelled by the character group π∗1(M) of the funda-
mental group of M.
We have to relate the sections σ of a line bundle η

with the elements ψ of L2(M, ν). The sections σ in η
form a complex vectorspace and, together with ν and
<, >, the ‘square’ integrable ones formaHilbert space
L2(η, <, >, ν). A dense domain of smooth sections in
η can be embedded in a dense domain D in L2(M, ν).
Hence we use this domain D to define differential op-
erators.
In general there are a large number of nonequiva-

lent differentiable structures on M × C which lead to
different dense domains in L2(M, ν). Hence one can
view the definition of differential operators with a do-
main problem.

II.

In part I., we explained how to introduce a differen-
tial structure. Now we refer again to the geometri-
cal roots to argue that our differential operators are
of finite order. Classical generators X of with sup-
port supp(X) moves with φX the characteristic func-
tion χ(B) resp. the support of f. After quantization
the situation should be analogous. This motivates a
locality condition for P(X)

supp(P(X)ψ)⊆ supp(ψ), ψ ∈ L2(M, ν). (12)

With Peetre’s theorem [14] this locality condition and
a differentiable structure -D(M × C) yields for P(X)
differential operators of finite order.
Collecting the results we have:
•ThequantummapPofmomentumobservable X

leads to differential operators of finite order on a do-
main in L2(M, ν) which is embedded on a domain in
L2(η, <, >, ν) spannedby square integrable sections of
a complex line bundle η on M withaHermiteanmetric
and a flat Hermitean connection.

3.2.3 Quantizations of the Borel kinematic
K(M)

With the properties for Q(f) and P(X) at the end of
section 3.2.1 and 3.2.2 we construct the quantummap
Q(M, K). The result reflects the classification of flat
complex line bundles with a Hermitean metric and a
Hermitean connection. A further classifying real num-
ber D is related to the global Lie-algebraic symmetry
K(M) (10) and the consistency of the partial realiza-
tion in Q(M, K).

Classification Theorem for Quantum Borel kine-
matics

Inequivalent irreducible quantum maps Q(M, K)
from K(M) to a cdi domain DQ in SA(L2(M, ν))

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Acta Polytechnica Vol. 50 No. 3/2010

Qα,D(M, K)= (Qα,D,Pα,D (13)

are labelled by two numbers α,D

α ∈ π∗1(M), D ∈ R. (14)

The domain DQ ⊂ L2(M, ν) is obtained through an
embedding in theHilbert space L2(η, <, >, ν) spanned
through square integrable sections of the complex line
bundle η on M with Hermitean metric and a flat Her-
mitean connection.
The α, D are quantum numbers in the sense of

Wigner. Quantizations Pα,D for different α and/or D
are unitary inequivalent. Quantizations Qα,D which
act on D as

Qα,D(f(m)))= f(m), f(m) ∈ C∞(M, R)

are independent of α and D. For Pα,D on D one ob-
tains a first order PDO through Lie-derivatives and
as the zero order part a smooth section of the endo-
morphism bundle which depends on D and α (see the
example in section 3.2.4 and [3]).
There are many applications. We refer to the list

in [4] and e.g. to quantizations on a trefoil mani-
fold [15] and on two and higher dimensional configu-
rationmanifolds for n identical particles [16] including
anyons.
Physical effects of quantummaps Qα,D with D =0

through quantum numbers α are experimentally ob-
served. However, for Qα,D with D �= 0 our deriva-
tion contains no information on the interpretation and
hints for its physical relevance; even in the case for
trivial α the physical meaning of D is unknown. Fur-
thermore Qα,D with D �=0 implies as a quantummap
of the kinematics no information on the time depen-
dence of the system, i.e. of its dynamics and further-
more no rule for the quantization (up to orderings)
of higher order (≥ 2) polynomials of momentum and
position observable.
A possible answer to these questions was proposed

by Jerry Goldin and HDD [1]. They introduced for
M = R3 a time dependence of ψ through particle con-
servation and obtainedwith Qα,D(R3, K) after ‘gauge
generalisation’ nonlinear Schrödinger equations with
nonlinear termsproportional to D. Theprocedure can
be generalizedto systemson M. These results indicate
a hidden nonlinear structure of {Qα,D(M, K3} which
will be explored in section 4.

3.2.4 Quantizations on M = R3

As an example for the classification theorem we con-

sider M = R3 with trivial π∗1(R
3) and X = −→g (x)

−→
∇ ,

f = f(x). We find for QD(R3, K) in D ∈ L2(R3,dx3)

QD(f) = f(x),

PD(X) =
1
i
−→g (x)

−→
∇ +

(
1
2i
+ D

)
div−→g (x). (15)

QD=0 ≡ Q0 is the canonical quantization. The maps
QD and QD

′
are unitarily inequivalent for D �= D′.

The expectation values can be scaled according to
their physical interpretation with automorphisms of
V ect0(M) � X �−→ aX and C∞(M, R) � f �−→ bf.

4 A nonlinear symmetry of{
QD

}
4.1 Nonlinear transformations and

operators

We use the above set
{
QD(R3, K)

}
to look for a ‘hid-

den non linear structure’. Because nonlinearities are
often a result of nonlinear transformations it is plausi-
ble to try physically motivated nonlinear transforma-
tions N of ψ ∈ L2(R3, d3x)

N:ψ −→ N[ψ] = N(ψ)ψ. (16)

We assume that N act asmultiplication operators and
depend only on ψ, i.e. not on derivatives of ψ and not
explicitly on x, t (domain and range questions are not
discussed in this paper). These transformation N[ψ]
could imply singularities e.g. in evolution equations,
which are not important here. In the case for multi-
valued N[ψ] one has to show that relevant results are
unique and independent from the choice of the repre-
sentatives of the ray {ψτ | ψ expiτ, τ real} which de-
scribe physical equivalent states.
To construct from esa operators A through N op-

erators AN, which may be linear or nonlinear, we
again use the flow model. The flow with generator
X corresponds after quantization to a strongly con-
tinuous one parameter unitary group U� with an esa
generator. This generator appears through a tan-

gent map
d
d�
(U�ψ)�=0 on a path {U�ψ, ∈ [−1,+1]} in

L2(R3,d3x). With this inmindwe construct AN from
a given esa A with V� = expi�A via a tangent map
N[V�ψ] with N on a corresponding nonlinear path

d
d�
(N[V�ψ])�=0 =

d
d�
(N[ψ + i�Aψ])�=0 ≡ (17)

iAN · N[ψ].

on a domain in which the limit exists. For differ-
ential operators AN this domain can be extended in
L2(R3,d3x). For some not esa linear A this construc-
tion is also possible.

4.2 Choice of nonlinear
transformations

Our aim is to determine a set N of nonlinear transfor-
mations such that QD,PD in (15)

QD = f, PD =
1
i
−→g ·

−→
∇ +

(
1
2i
+ D

)
div−→g

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Acta Polytechnica Vol. 50 No. 3/2010

are connected for different D through a tangent map
with N ∈ N

QD
′
= QDN with Q

D′ =QDN , P
D′ =PDN . (18)

An evaluation of these conditions is indeed possi-
ble; the calculation is straightforward. With a (non
unique) polar decomposition ψ = RexpiS we find for
the nonlinear transformations N

N(γ,Λ)ψ =expi(γ lnR +(Λ−1)S) · ψ. (19)

They build a nonlinear representation NG =
{N(γ,Λ)}of twoparameter (γ,Λ)group G. Thegroup
NG was derived in a different context and with other
assumptions in [17]. For the corresponding tangent
maps we find

QD = Q0N(γ,Λ) with γ =2D, Λ.arbitrary

QD
′
= QDN(γ,Λ) with γ =2(D

′ − D), Λ=1.(20)

Hence our search for a hidden symmetry was success-
ful. The result implies:
• Quantizations QD and QD

′
are inequivalent in

respect to the group of unitary transformations but
‘equivalent’ in respect to a nonlinear realization NG
of a matrix group G with one and for D = 0 and
for D′ �= 0 with two real parameters. The quantum
number D leads to a nonlinear representation of a G
symmetry of

{
QD(K(R3))

}
.

4.3 Nonlinear quantizations and
symmetries

We now extend our construction from linear first and
zero order differential operators in Q(R3, K) to esa
higher order differential operators operators in poly-
nomials of canonically quantized observables

D ∈ Pesa(Q0(f),P0(X)).

An application of tangent maps leads to

N(γ,Λ): D �→ DN(γ,Λ) (21)

i.e. to a two parameter set of nonlinear differential op-
erators DN(γ,Λ). This nonlinear ‘extension’ of canoni-
cally quantized polynomialsD of classical observables
through tangent maps with NG is an attempt for a
formal path to nonlinear ‘extensions’ of quantum me-
chanics. However, an interpretation of nonlinear op-
erators and nonlinear evolutions as in (linear) quan-
tum theory is not possible. Results from a nonlinear
theory can be interpreted in some approximation as
in the linear case, e.g. the eigenvalues of a nonlin-
ear Schrödinger equation (22) (see [18]). A complete
mathematical framework and convincing physical in-
terpretations for a nonlinear ‘extension’ is not known.
With (21) one obtains an already mentioned non-

linear realization of G(3) with linear generators of the

Galilei-algebrawith the exception of the freeHamilto-
nian which appears as nonlinear operator. Further-
more one gets with N ∈ NG from a given linear
Schrödinger equation a nonlinear one with nonlinear
part F [ψ] which depends on γ,Λ. This nonlinear
equation was generalized in [19] to a nonlinearisable
Schrödinger equation (known as the Doebner Goldin
equation) with real coefficients D, λ1, . . . , λ5

ih̄∂t = (−h̄2/2m%+ F [ψ])ψ

F [ψ] =
ih̄D

2
%ρ
ρ
+ h̄D {λ1R1 + . . . + λ5R5} (22)

with

ρ = ψψ, J =
h̄

2mi

{
ψ∇ψ −∇ψψ

}
R1[ψ] =

∇· J
ρ

, R2[ψ] =
%ρ
ρ

, R3[ψ] =
J2

ρ2

R4[ψ] =
J ·∇ρ

ρ2
, R5[ψ] =

∇ρ ·∇ρ
ρ2

5 Conclusion
We announced in the preview that we would re-
late global and nonlinear structures of quantum maps
Q(M, K) for the Borel kinematics K(M) of nonrela-
tivistic systems on M without internal degrees of free-
domand external fieldswith application forQ(R3, K).
• Inequivalent quantum maps

{
Qα,D(M, K)

}
, la-

belled through quantum numbers α, D in the sense of
Wigner which reflect topological and global properties
of M and K(M). Experiments to measure effects in
Qα,D are known for D =0.
• For M = R3 and D �= 0 these global structures

imply that
{
QD(R3, K)

}
carries a nonlinear represen-

tations NG of a 2×2 matrix group G.
• For esa differential operators D of order ≥ 2

in the polynomial set Pesa(Q0(f),P0(X)) one obtains
nonlinear differential operators DN through tangent
maps with N ∈ NG. Nonlinear versions of symmetry
and dynamical symmetry algebras are available. Our
constructionmaybe viewedas part of a path to a non-
linear extension of quantum mechanics. This may be
relevant in the case that precision experiments show a
nonlinear character and corresponding nonlinear evo-
lutions based on a global character of M and K(M).

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H.-D. Doebner
E-mail: asi@pt.tu-clausthal.de, doebner@t-online.de
Technical University of Clausthal
Institute for Energy Research and Physical Tecnology

81