Acta Polytechnica Vol. 51 No. 4/2011

Path Integrals for (Complex) Classical and Quantum Mechanics

R. J. Rivers

Abstract

An analysis of classical mechanics in a complex extension of phase space shows that a particle in such a space can behave
in a way redolent of quantum mechanics; additional dimensions permit ‘tunnelling’ without recourse to instantons and
time/energy uncertainties exist. In practice, ‘classical’ particle trajectories with additional degrees of freedom arise in
several different formulations of quantum mechanics. In this talk we compare the extended phase space of the closed
time-path formalism with that of complex classical mechanics, to suggest that h̄ has a role in our understanding of the
latter. However, differences in the way that trajectories are used make a deeper comparison problematical. We conclude
with some thoughts on quantisation as dimensional reduction.

Keywords: phase space, path integral, complex classical physics.

1 Introduction
It has been argued recently, in a series of papers by
Carl Bender and collaborators [1–3], that classical
mechanics in a complex extension of phase space has
some attributes of quantum mechanics. With these
additional dimensions, particles can negotiate oth-
erwise impenetrable classical hills, enabling them to
move from one potential well to another as an al-
ternative to quantum tunnelling. However, the dis-
cussion has now gone beyond the mere cataloguing
of these qualitative similarities, to suggesting that
we are seeing behaviour that can approximate the
probabilistic results of quantum mechanics quantita-
tively [3]. Although the emphasis has been on mim-
icking h̄-independent probability ratios, for this pro-
cedure to be sensible h̄ must be implicit in the anal-
ysis, since the formalism of complex classical me-
chanics cannot, of itself, distinguish between quan-
tum (action O(h̄)) systems and classical systems (ac-
tion O(h̄0)). In fact, should there be any relation-
ship between complex classical mechanics and quan-
tum mechanics, there is a strong hint as to how
this must happen, since the tunnelling complies with
time/energy uncertainty relations [1], albeit with no
h̄ visible.

On the other hand, if we adopt the contrary
position of extending (real) classical behaviour to
quantum mechanical behaviour, it is well-known that
there are several different ways in which classical or
‘quasi-classical’ paths can arise in the formulation of
quantum dynamics, for which the role of h̄ is clear. In
particular, we are interested in what we might term a
Moyal path integral approach which (in co-ordinate
space) reduces to the Feynman-Vernon closed time-
path approach [4] for the evolution of density matri-
ces, familiar in the analysis of decoherence. In fact, it

was the use of classical trajectories by the author [5,6]
to approximate the evolution of the quantum den-
sity matrix that provoked this talk. In this com-
mentary we shall contrast the quasi-classical paths of
this Moyal formalism to the classical paths in com-
plex phase-space, in each case the solutions to a con-
strained Hamiltonian system.

For reasons that will rapidly become clear, we
shall initially restrict ourselves to Hermitian Hamil-
tonians, even though part of the original motivation
for considering complex phase space was to accom-
modate pseudo-Hermitian Hamiltonians which, by
virtue of PT-symmetry, had real spectra. There is a
huge literature on this field, and we would cite [7–9]
as exemplary. In the next two sections we introduce
path integrals for real classical mechanics, as devel-
oped by Gozzi over several years [10–12] and show
how the Moyal path integrals for quantum mechani-
cal systems evolve naturally from them, as originally
proposed by Marinov [13] and Gozzi [14, 15]. We
show that the dynamics of the quasi-classical trajec-
tories that provide the mean-field approximation to
this quantum system has the same symplectic struc-
ture as the trajectories of complex classical mechanics
shown by Smilga [18]. Insofar as the two approaches
have some similarities we may hope to impute h̄ be-
haviour to complex classical mechanics. Insofar as
they have differences it might be thought that, the
better that Moyal paths represent quantum mechan-
ics, to which they are an identifiable approximation,
the more poorly the paths from CCM will do so. In
fact, as we shall see, the situation is somewhat more
complicated. Even prior to papers [13] and [14, 15]
there was an extensive literature on the role on clas-
sical and quasi-classical paths in quantum mechanics
(e.g. [16]) that has been developed since. Given the
brevity and simplicity of our observations it is suffi-

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cient, in the context of this paper, to cite just [17]
and the references therein for interested readers.

We conclude with some thoughts on quantisation
as dimensional reduction.

2 Phase space path integrals
for (real) classical
mechanics

We restrict ourselves to a single particle, mass m,
moving in phase space ϕ = (x, p), under the Hamil-
tonian

Hcl(ϕ) := Hcl(x, p) =
p2

2m
+ V (x).

The classical solutions ϕcl satisfy Hamilton’s equa-
tions

ϕ̇a − ωab∂bHcl = 0, (2.1)
where ωab = iσ2 and ∂a = ∂/∂ϕ

a.
Classical phase-space densities ρcl(ϕ, t) evolve as

ρcl(ϕf , tf ) = (2.2)∫
Dϕa Kcl(ϕf , tf |ϕi, ti)ρcl(ϕi, ti)

where the kernel Kcl(ϕf , tf |ϕi, ti) is restricted to the
classical paths of (2.1),

Kcl(ϕf , tf |ϕi, ti) =
∫ ϕf

ϕi

Dϕa δ[ϕa − ϕacl] = (2.3)∫ ϕf
ϕi

Dϕa δ[ϕ̇a − ωab∂bHcl]

The second equality of (2.3) is a consequence of the
incompressibility of phase space.

We repeat the earlier analysis of Gozzi [10] in
doubling phase space through the functional Fourier
transform of the delta functional:

Kcl(ϕf , tf |ϕi, ti) = (2.4)∫ ϕf
ϕi

DϕaDΠa exp
{

i

∫ tf
ti

dt Πa(ϕ̇
a − ωab∂bHcl)

}
If 〈. . .〉 denotes averaging with respect to the (nor-
malised) path integral then, on time-splitting, we see
that the Πa are conjugate to the ϕ

a, with equal-time
commutation relations

〈[ϕa, Πb]〉 = iδab (2.5)

That is, in (2.4) we have the path integral reali-
sation of the canonical Koopman — von Neumann
(KvN) Hilbert space description of classical mechan-
ics [19, 20]. This becomes clearer if we rewrite
Kcl(ϕf , tf |ϕi, ti) as

Kcl(ϕf , tf |ϕi, ti) = (2.6)∫ ϕf
ϕi

DϕaDΠa exp
{

iScl[ϕ, Π]

}

where

Scl[ϕ, Π] =
∫ tf

ti

dt Lcl(ϕ, Π) (2.7)

with

Lcl(ϕ, Π) = ϕ̇aΠb − Hcl(ϕ, Π).

The Hamiltonian Hcl(φ, Π),

Hcl(ϕ, Π) = Πaωab∂bHcl. (2.8)

is no more than the Liouville operator (up to a fac-
tor of i), as follows immediately if we adopt the ϕ-
representation Πa = −i∂a in Hcl. The commutators
with Hcl(ϕ, Π) (in the sense of (2.5)) determine the
evolution of (ϕ, Π), but it is more convenient to think
of these solutions as just the solutions (ϕcl, Πcl) to
δScl = 0.

3 Phase space path integrals
for quantum mechanics

To explore the role of semi-classical paths in quan-
tum mechanics we stay as close to the classical for-
malism of the previous section as possible. The clas-
sical phase-space density ρcl(ϕ, t) is replaced by the
Wigner function ρW (ϕ, t)

ρW (x, p, t) = (3.9)
1

πh̄

∫
dy 〈x − y|ρ̂(t)|x + y〉e2ipy/h̄,

which, although not strictly a density, reduces to it
in the h̄ → 0 limit.

Its evolution equation

ρW (ϕf , tf ) = (3.10)∫
Dϕi Kqu(ϕf , tf |ϕi, ti)ρW (ϕi, ti)

is determined by the kernel Kqu, which we define on
the same extended phase-space as

Kqu(ϕf , tf |ϕi, ti) = (3.11)∫ ϕf
ϕi

DϕaDΠa exp
{

iSqu[ϕ, Π]

}
,

with

Squ(ϕ, Π) =
∫ tf

ti

dt Lqu(ϕ, Π) = (3.12)∫ tf
ti

dt (ϕ̇aΠb − Hqu(ϕ, Π))

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The quantum Hamiltonian Hqu(φ, Π) is a Planck-
ian finite-difference discretisation of Hcl(φ, Π), to
which it reduces as h̄ → 0. Several choices are possi-
ble. For the reasons given in [14, 15] we follow these
authors in taking

Hqu(ϕ, Π) = −
1

2h̄
[Hcl(ϕ

a + h̄ωabΠb) −

Hcl(ϕ
a − h̄ωabΠb)]. (3.13)

Although we appreciate that the paths themselves
can be less important than the ways in which they
are put together, what singles out complex classi-
cal mechanics (CCM) is the importance attached to
individual paths in tracking their times of passage
through the important regions of the complexified
classical potential landscape. A priori, we take the
same stance here, in assuming that Kqu is dominated
by solutions to

δSqu[ϕ, Π] = 0. (3.14)

That is, we treat Squ[ϕ, Π] as a quasi-classical the-
ory in its own right, which we shall term mean-field
quantum mechanics (MFQM). Since Squ = O(h̄

0),
(3.14) is a stationary phase approximation with no
small parameter, and therefore to be taken circum-
spectly.

In what follows we compare MFQM to CCM. Al-
though they do not match they have suggestive sim-
ilarities. To cast Squ in a more familiar form, we re-
produce Marinov [13] by introducing new phase space
variables:

ξa := h̄ωabΠb. (3.15)

Kqu then takes the integral form

Kqu(ϕf , tf |ϕi, ti) =∫ φf
φi

DϕaDξa exp
{

i

h̄
SM [φ, ξ]

}
, (3.16)

where

SM [ϕ, ξ] =
∫ tf

ti

dt LM (φ, ξ) (3.17)

with

LM (φ, ξ) = ϕ̇aωabξb + (3.18)
1
2

[Hcl(ϕ + ξ) − Hcl(ϕ − ξ)]

We stress again that the formalism of (3.16) is
misleading, in that it suggests that the station-
ary phase approximation is also a small-h̄ result,
whereas SM is O(h̄). Despite that, there has
been considerable work that successfully utilises the
stationary-phase solutions [see [17] and applications
cited therein].

3.1 Structure of MFQM

Let us define ξ1 := y, ξ2 := q and

H± :=
1
2

[Hcl(ϕ + ξ) ± Hcl(ϕ − ξ)] (3.19)

with ϕ ± ξ = (x ± y, p ± q). We rearrange the original
extended phase-space (ϕ, Π) into the 4D phase space
X:

X1 := x, X2 := y, X3 := p, X4 = q.

On X we introduce the Poisson bracket {{·, ·}}:

{{A, B}} := Ωab ∂aA ∂bB, (3.20)

where

Ω =

(
0 I

−I 0

)
. (3.21)

Hamilton’s equations, which reduce to δSqu = 0, then
take the form

Ẋ a = {{X a, H+(X)}} = Ωab∂bH+(X). (3.22)

Since H+ and H− are related by ∂aH
− =

Γba∂bH
+, where

Γ =

(
σ1 0

0 σ1

)
(3.23)

it follows that

{{H−, H+}} = ΩacΓbc ∂aH
+ ∂bH

+ = 0, (3.24)

since ΩΓ is antisymmetric. H− = const. is a first
class constraint upon the effective classical theory.

We note that there is an equivalence between H+

and H− in that, with respect to a slightly different
symplectic matrix Ω′, we could equally derive the
equations of motion from H− as

Ẋ a = {{X a, H−(X)}}′ = Ω′ab∂bH−(X). (3.25)

This is more in accord with the closed timepath for-
malism (CTP), defined in the extended (x, y) coordi-
nate space, for which H− is the relevant Hamiltonian
prior to momentum integration. We shall not pursue
this further.

The classical limit is straightforward. Remem-
ber that y = h̄ȳ, q = h̄q̄ where ȳ, q̄ are the O(h̄0)
KvN conjugate variables to p and x. As h̄ → 0 then
y, q → 0, as does

H− = O(h̄) → 0. (3.26)

At the same time we get the contraction H+ → Hcl,
Ω → ω.

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4 Complex Classical
Mechanics (CCM)

We now consider complex phase space, repeating the
analysis of Smilga [18], taking the complex extension
of the real phase-space as

x → Z = x + iy, p → P = p − iq. (4.27)

The Hamiltonian Hcl is decomposed as

Hcl(p, x) → (4.28)
Hcl(P, Z) = HR(P, Z) + iHI (P, Z),

where

HR =
1
2

[Hcl(x + iy, p − iq) + (4.29)

Hcl(x − iy, p + iq)],

HI =
1
2i

[Hcl(x + iy, p − iq) − (4.30)

Hcl(x − iy, p + iq)].

To see the symplectic structure of CCM we label
the phase-space variables X a as before:

X1 := x, X2 := y, X3 := p, X4 = q. (4.31)

and introduce an identical Poisson bracket {{·, ·}} to
(3.20):

{{A, B}} := Ωab ∂aA ∂bB, (4.32)

for the same Ω. Hamilton’s equations are now [18]

Ẋ a = {{X a, HR}} = Ωab∂bHR(X). (4.33)

HR and HI are related by Cauchy-Reimann as

∂aHI = Λ
b
a∂bHR, (4.34)

where

Λ =

(
−iσ2 0

0 iσ2

)
. (4.35)

It follows that

{{HI , HR}} = (ΩΛ)ab ∂aHR ∂bHR = 0, (4.36)

since ΩΛ is also antisymmetric. That is, as before we
have a constrained system with first-order constraint
HI = constant.

5 CCM v. MFQM

There are obvious similarities between the two ap-
proaches, a consequence of the identities

H+(x, iy, p, −iq) = HR(x, y, p, q)
H−(x, iy, p, −iq) = iHI (x, y, p, q). (5.37)

Since
{{y, q}} = {{iy, −iq}}

we can see why the symplectic structure remains un-
changed.

The similarity between the two formalisms is very
apparent for the SHO, with Hamiltonian

Hcl =
1
2

(p2 + x2).

In the x−y plane we find identical solutions for x and
y in both CCM and MFQM. These are tilted ellipses

x = A sin(t + α1), y = B sin(t + α2),

leading to identical H− and HI ,

H− = HI = AB cos(α1 − α2)

This suggests a possible role for h̄ in CCM for this and
other potentials. We remember that y = h̄ȳ, q = h̄q̄
in MFQM, which encourages us to take y = h̄ȳ, q =
h̄q̄ in CCM. Then
• CCM can now distinguish between ‘large’ and

‘small’ systems. The conventional classical limit
is simply understood as the recovery of real
phase space.

• With 0mE now O(h̄) the empirical CCM tun-
nelling observation 0mE Δt ≈ const. is un-
derstood as the quantum uncertainty relation
ΔEΔt = O(h̄).

• The CCM tunnelling results in [3] are un-
derstood as anomalous behaviour in the limit
0mE → 0. We now understand this as the fa-
miliar small h̄ behaviour when looking for the
persistence of ‘quantum’ effects.

Looking at more general potentials, even in [13]
and [14] it was appreciated that the extra dimensions
permitted tunnelling without instantons. However,
this should not blind us to strong differences between
the two formalisms, for which the factors of i are cru-
cial. Most importantly, the constant energy surfaces
of MFQM are bounded, whereas those of CCM are
unbounded. As a result, individual particle trajecto-
ries in CCM go to infinite distances in the x−y plane
(and back) in finite time, whereas those of MFQM are
always bounded.

To see the effects of this boundedness, it is con-
venient (with the former) to work with z± = ϕ ± ξ,
since Hcl(z±) are individually conserved:

ża± = ω
ab ∂Hcl(z±)

∂zb±

We can think of the classical paths for z± as the tips
of chords of length O(h̄) whose midpoints are quan-
tum paths, only constrained by boundary conditions.
In the Lagrangian formalism (on integrating out p, q)
this is the familiar closed time-path approach.

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For example, now consider ‘tunnelling’ in a
double-well potential with binding energy E0. In
CCM we take HR = En < E0 to match the energy of
a bound state of the potential and fix HI = 0mE =
ΔE �= 0. If, for example, we then consider paths with
starting points y = 0, x = x0 the particle flips from
well to well in a ‘symmetric’ way using the additional
dimensions [3]. If we now measure the ratio of the
time the particle spends in each well as ΔE → 0 we
can compare this with the QM results obtained from
the wavefunctions as HI = 0mE → 0. See [3] for
details. While not being compelling, the results are
certainly interesting.

However, the situation is very different in MFQM
because of the boundedness of the constant energy
surfaces. We now have

H+ =
1
2

[Hcl(z+) + Hcl(z−)].

As shown in [17], to proceed we choose H+ = E < E0
with Hcl(z+) > E0 and Hcl(z−) < E0 (or v.v) and
fix H− = ΔE �= 0. That is, one end of the chord is
trapped in a well, while the other end is free. The
initial conditions mean that the particle flips from
well to well ‘asymmetrically’ using additional dimen-
sions. For this reason it is not sensible to attempt to
measure the relative time the particle spends in each
well as ΔE → 0. These trajectories make fundamen-
tally clear a profound difference between CCM and
MFQM that lies at the heart of quantum mechan-
ics. For all the qualitative similarities between CCM
and quantum mechanics, the defining ingredient of
the latter is interference. This is immediately clear
from the evolution equation (3.11) for real density
matrices, which demands that Kqu be real. At the
very least, if (ϕcl, ξcl) are solutions to (4.33) then so
are (ϕcl, −ξcl) and both solutions have to be com-
bined (z+ ↔ z−) in (3.11), with the appropriate de-
terminant of small fluctuations around the classical
solutions.

This is in contrast to the Hamiltonian formula-
tion of classical mechanics of Section 2 for which
there are potentially more observables, i.e. Hermi-
tian functions of ϕ and Π, than in the standard ap-
proach to (real) classical mechanics. Because of the
non-commutativity of ϕ and Π, as shown in (2.5),
interference looks to be possible, although we know
this is not the case. In fact, not all these extra ob-
servables are invariant under a set of universal local
symmetries which appear once the formalism is ex-
tended to differential forms on phase space [21] and,
because of this, have to be removed. As Gozzi has
shown, this makes the superposition of states in (real)
CM impossible [22]. Whether this is equally true for
complex classical mechanics is unclear, but we would
be surprised if it were not. Unfortunately, this makes
a direct comparison between the approaches impos-
sible. Thus, even in a pragmatic sense, we can’t use

the virtues of the one to have implications for the
other.

In fact, we need more than interference between
quasi-classical paths in MFQM, as can be seen from

the simple V (x) = −
1
2
x2 potential. Quasi-classical

solutions in MFQM show that the particle rebounds
for E < 0, i.e. there is no ‘tunnelling’ despite the
extra dimension. Nonetheless, tunnelling happens in
MFQM because of state preparation [23], whereby
the tail of the wavefunction crosses the x − p separa-
trix and is stretched to the other side of the potential
hill.

We conclude with a comment on the first order
constraints H− = 0 and HI = 0 that the formalisms
possess. To accommodate them fully requires gauge
fixing in the path integrals (or a change of bracket).
At our simple level of comparison this is unnecessary,
but see Smilga [18] for a detailed discussion of gauge-
fixing for CCM.

6 Quantisation as dimensional
reduction

So far we have compared the classical trajectories of
particles in complex phase space with the classical
trajectories z± of the ends of the chords whose mid-
points are the stationary phase solutions that define
what we have called mean-field quantum mechanics.
The doubling of the degrees of freedom by having
to take account of two chord ends has its counter-
part in the doubling of degrees of freedom by making
phase space complex. Since the chords are of length
O(h̄) we recover real classical mechanics from mean-
field quantum mechanics trivially and, if the com-
plex phase-space coordinates are, equally, O(h̄), re-
cover real classical mechanics from complex classical
mechanics at the same time. However, the real sig-
nificance of the doubling of degrees of freedom in the
chords is that these new coordinates are the quantum
generalisations of the classical ‘momenta’ conjugate
to the real phase space variables in the KvN Hilbert
space formalism of classical mechanics.

This suggests an entirely different approach to
‘deriving’ quantum mechanics from classical mechan-
ics that we sketch below. More details will be
presented elsewhere. It relies on the extension of
the formalism of Section 2 to differential forms by
Gozzi [10], that we have already alluded to above.
For the moment we stay firmly with classical me-
chanics in real phase space. The rightmost integral
of (2.3) contains the Jacobian

J := det(δab ∂t − ω
ac∂c∂bH) (6.38)

which we have set to unity, as a consequence of the
incompressibility of phase space. However, we can

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equally express it in terms of 2 + 2 ghost-field Grass-
mann variables (ca, c̄a) as [10]

J =
∫

Dc̄aDca · (6.39)

exp

[
−
∫

dtc̄a[δ
a
b ∂t − ω

ac∂c∂bH]c
b

]
.

Inserting this in the integrand of the rightmost
path integral in (2.3) enables us to write the kernel
Kcl as

Kcl =
∫

DX aDΠaDc̄aDca exp
[
i

∫
dt L̄cl

]
(6.40)

where we have dropped explicit mention of boundary
conditions to simplify the notation. In (6.40) Lcl of
(2.8) is replaced by

L̄cl = Πa[Ẋ a − ωab∂bH] + ic̄a[δab ∂t − ω
ac∂c∂bH]c

b

= ΠaẊ
a + ic̄aċ

a − H̄cl, (6.41)

where the Hamiltonian associated with L̄ of (6.41) is
now

H̄cl = Πaωab∂bH + ic̄aωac∂c∂bHcb. (6.42)

The equations of motion that follow from L̃cl show
that the (ca, c̄a) are conjugate Jacobi variables in the
sense of (2.5) and that H̄cl is the Lie derivative of
Hamiltonian flow associated with Hcl [10].

All that matters for this discussion is that, on
introducing Grassmann partners (θ, θ̄) to time t,
we can construct superspace phase space variables
Φ = (X, P )

Φa(t, θ, θ̄) := ϕa(t) + θca(t) + (6.43)

θ̄c̄a(t) + iθ̄θωabΠb

We note that θ̄θ has the dimensions of inverse action.
It follows [10] that

i

∫
dθ dθ̄ Hcl(Φ) = Hcl(ϕ, Π). (6.44)

Similarly,

i

∫
dt dθ dθ̄ Lcl(Φ) =

∫
dt Lcl(ϕ, Π), (6.45)

where
Lcl = pẋ − Hcl. (6.46)

The theory possesses BRS invariance. We re-
trieve quantum mechanics from classical mechanics
by making the dimensional reduction [24]

ih̄

∫
dt dθ dθ̄ →

∫
dt (6.47)

in superspace, together with (X, P ) → (x, p) in phase
space. See also [25] for the relationship of this ap-
proach to ‘t Hooft’s derivation of quantum from clas-
sical physics [26, 27].

This analysis permits a natural extension to com-
plex phase space, with co-ordinates X a of (4.31).
The first step is to double this already doubled phase
space by introducing four conjugate variables Π̄a, to
be supplemented by (now 4 + 4) Grassmann vari-
ables (ca, c̄a). We then look for BRS symmetry in
this extended space. A superspace realisation of
this extended theory is possible. It is then inter-
esting to look at its dimensional reduction, not yet
attempted. If the outcome is quantum mechanics, it
will be quantum mechanics defined on complex phase
space, i.e. we shall be looking at probability densi-
ties in the complex plane [28]. This is to be distin-
guished from the results of [3], in which comparison
is made between CCM and quantum mechanics in
the real plane. However, it will allow for a discussion
of pseudo-Hermitian Hamiltonians, whose quantum
mechanics has already been described in detail [7–9]
and with which much of the discussion of [18] was
concerned.

7 Conclusions
Our conclusions derived from these explorations of
path integrals are somewhat schizophrenic. On the
one hand, if we take the behaviour of particles in
complex classical mechanics (CCM) as really reflect-
ing attributes of quantum mechanics (QM), then for-
mal similarities between CCM and mean-field quan-
tum mechanics (MFQM) suggest that the complex
dimensions of CCM should be taken as O(h̄). This
resolves several issues with CCM, such as how clas-
sical mechanics distinguishes between classical and
quantum systems, and how to interpret uncertainty
relations.

On the other hand, the differences between CCM
and MFQM are at least as important as the simi-
larities. In particular, the boundedness of the en-
ergy surfaces of the latter (in comparison to the un-
bounded nature of those of the former) mean that
the trajectories are very different, even though each
permits tunnelling without instantons and we know
that, in many circumstances, MFQM gives a reliable
description of quantum mechanical particles. In par-
ticular, for MFQM quantum superposition is a nec-
essary ingredient, whereas (for real phase space at
least) superposition plays no role in the path inte-
grals of classical mechanics.

Much of the original work on CCM was concerned
with comparing its results to quantum mechanics in
the real plane, e.g. [3]. As an alternative, we have
raised the possibility of looking for supersymmetric
realisations of CCM, with the potential of getting
quantum mechanics in complex phase space. This
will be pursued elsewhere.

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Acknowledgement

We acknowledge stimulating discussions with Ennio
Gozzi and thank him for access to unpublished notes
on ‘tunnelling without instantons’ (December, 1995).
We also thank Daniel Hook for empirically demon-
strating the nature of ‘tunnelling’ paths in MFQM.

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R. J. Rivers
E-mail: r.rivers@imperial.ac.uk
Physics Department
Imperial College London
London SW7 2AZ, UK

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