Nonequilibrium action principles for quasi-field
Quasiparticle descriptions of nonequilibrium physics hold the promise of tracking the relevant degrees of freedom in calculations of statistical systems, as well as providing an important formal perspective on quantum field theories away from equilibrium. A formulation of nonequilibrium, quasiparti...
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nasplib_isofts_kiev_ua-123456789-1210402025-02-09T17:00:48Z Nonequilibrium action principles for quasi-field Нерівноважний принцип дії для квазіполів Burgess, M. Quasiparticle descriptions of nonequilibrium physics hold the promise of tracking the relevant degrees of freedom in calculations of statistical systems, as well as providing an important formal perspective on quantum field theories away from equilibrium. A formulation of nonequilibrium, quasiparticle field theory is presented here. The Schwinger closed time path generating functional and generalized sources are used to develop schematic field theories which can be adapted to real problems. The importance of the source method for effective nonequilibrium theories is argued and the physical interpretation of the formalism is discussed, including its possible applicability to biological dynamics. Квазічастинкові описи нерівноважної фізики дозволяють простежувати відповідні ступені вільности в обчисленнях статистичних систем і забезпечують важливий формальний підхід до квантової теорії поля поза рівновагою. Тут запропоновано формулювання нерівноважної квазічастинкової теорії поля. Для розвитку схематичних теорій поля, здатних описувати реальні системи, використовуються швінґерівський твірний функціонал за шляхами, замкненими в часі, і узагальнені джерела. Обґрунтовується важливість методу джерел для ефективних нерівноважних теорій, а також його застосовність до біологічної динаміки. 2000 Article Nonequilibrium action principles for quasi-field / M. Burgess // Condensed Matter Physics. — 2000. — Т. 3, № 1(21). — С. 35-50. — Бібліогр.: 23 назв. — англ. 1607-324X DOI:10.5488/CMP.3.1.35 PACS: 03.70.+k, 05.30.-d, 05.70.Ln https://nasplib.isofts.kiev.ua/handle/123456789/121040 en Condensed Matter Physics application/pdf Інститут фізики конденсованих систем НАН України |
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Quasiparticle descriptions of nonequilibrium physics hold the promise of
tracking the relevant degrees of freedom in calculations of statistical systems, as well as providing an important formal perspective on quantum
field theories away from equilibrium. A formulation of nonequilibrium, quasiparticle field theory is presented here. The Schwinger closed time path
generating functional and generalized sources are used to develop schematic field theories which can be adapted to real problems. The importance
of the source method for effective nonequilibrium theories is argued and the
physical interpretation of the formalism is discussed, including its possible
applicability to biological dynamics. |
| format |
Article |
| author |
Burgess, M. |
| spellingShingle |
Burgess, M. Nonequilibrium action principles for quasi-field Condensed Matter Physics |
| author_facet |
Burgess, M. |
| author_sort |
Burgess, M. |
| title |
Nonequilibrium action principles for quasi-field |
| title_short |
Nonequilibrium action principles for quasi-field |
| title_full |
Nonequilibrium action principles for quasi-field |
| title_fullStr |
Nonequilibrium action principles for quasi-field |
| title_full_unstemmed |
Nonequilibrium action principles for quasi-field |
| title_sort |
nonequilibrium action principles for quasi-field |
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Інститут фізики конденсованих систем НАН України |
| publishDate |
2000 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/121040 |
| citation_txt |
Nonequilibrium action principles for quasi-field / M. Burgess // Condensed Matter Physics. — 2000. — Т. 3, № 1(21). — С. 35-50. — Бібліогр.: 23 назв. — англ. |
| series |
Condensed Matter Physics |
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2025-11-28T07:30:24Z |
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2025-11-28T07:30:24Z |
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| fulltext |
Condensed Matter Physics, 2000, Vol. 3, No. 1(21), pp. 35–50
Nonequilibrium action principles for
quasi-fields ∗
M.Burgess
Oslo College, Cort Adelers gate 30, 0254 Oslo, Norway
Received June 1, 1999
Quasiparticle descriptions of nonequilibrium physics hold the promise of
tracking the relevant degrees of freedom in calculations of statistical sys-
tems, as well as providing an important formal perspective on quantum
field theories away from equilibrium. A formulation of nonequilibrium, quasi-
particle field theory is presented here. The Schwinger closed time path
generating functional and generalized sources are used to develop sche-
matic field theories which can be adapted to real problems. The importance
of the source method for effective nonequilibrium theories is argued and the
physical interpretation of the formalism is discussed, including its possible
applicability to biological dynamics.
Key words: nonequilibrium field theory
PACS: 03.70.+k, 05.30.-d, 05.70.Ln
1. Introduction
Scientific studies are usually performed for one of two reasons: either to directly
address and explain a phenomenon or to indirectly explore the meaning of a method-
ology. In the latter, one hopes to better understand how dynamical formulations of
physics fit into the greater scheme of things – about universal qualities and struc-
ture. It is from both these perspectives that it is interesting to approach the problem
of quasi-particles.
Quasi-particles are “dressed excitations” of a quantum field. They are dynam-
ical entities which drag with them a cloud of correlating interactions from their
surrounding environment. In large statistical systems all particles are surrounded
by a bath or reservoir of average environmental behaviour, whether it is a heat
bath, a phenomenological source or an explicitly interacting quantum field. Parti-
cles’ surroundings modify their behaviour in complex ways and often it is possible
to find a simpler effective description in terms of a new set of dynamical objects or
quasi-particles which absorb part of that complexity into redefinitions of the basic
variables.
∗This is an invited paper to the special issue on the problems of thermofield dynamics.
c© M.Burgess 35
M.Burgess
The fact that quasi-particle descriptions track the important changes in dynami-
cal objects and hide the less obvious details of the interaction clouds is an attractive
idea: it suggests that conceptual and perhaps calculational simplifications can be
made essentially by a change of variable. In practice the use of quasi-particle de-
scriptions may prove too difficult to use as a calculational scheme, since they involve
hierarchies of non-local correlations and these are notoriously difficult to work with,
as we know from gauge theories. On the other hand a deeper understanding of the
way in which quasiparticle formalism works can help us to build effective theories
whose local remnants yield tractable problems at sufficiently long wavelengths and
might also be important in revealing universal aspects. Some progress has been made
here using rather different approaches [1–3].
Nonequilibrium field theory presents its own special problems to effective theory
building. By its very nature, the effective dynamical variables of a nonequilibrium
field theory change with time and often also from place to place, so the identification
of appropriate variables which track these changes is a difficult problem. In fact the
problem of an effective description of spacetime varying quasi-particles is in many
respects identical to the study of gauge theories. In a gauge theory one deals with
phases which vary in space and time; the effects of those phases can also be repre-
sented through a transformation in terms of a vector potential which provides an
‘environmental background’ for ‘particles’ or excitations of a field. In a nonequilib-
rium field theory one is interested in statistical environmental changes which vary
in space and time. It is perhaps not surprising then that these can be represented
in terms of a ‘statistical vector potential’.
The primary purpose of this review is to demonstrate that the formal structure
of nonequilibrium field theory has much in common with covariant field theories,
but in a wider context, i.e. to reassert the importance of covariant formulation and
hidden symmetry.
2. Inhomogeneous field theory
Nonequilibrium systems remember their history: they are not translationally
invariant in time and often not translationally invariant in space. A formalism which
captures the long-term changes in the system is required. This will typically involve
two scales: one scale at which long term average changes take place and another
over which fluctuations occur. To measure a scale we need two points so we expect
two point nonequilibrium Green functions to have special properties, so we look for
a formalism in which these scales can be separated from one another.
Between any two space-time points x and x′ it is useful to parameterize functions
in terms of ‘rotated’ variables [1]:
x̃ = (x− x′), x =
1
2
(x+ x′). (1)
The odd variables x̃ (the variable conjugate to the momentum) characterize ‘trans-
lational invariance’ while the even variables x represent the opposite of this: inho-
mogeneity.
36
Nonequilibrium quasiparticles
Many problems in nonequilibrium field theory can be addressed with the help
of the closed time path (CTP) generating functional [4–6]. The CTP is a field the-
oretical prescription for deriving expectation values of physical quantities, given a
description of the state of the field at some time in the past. The generating func-
tional requires an artificial duplicity in the field, so the closed-time path action is
described as a two-component field φA, where A = +,−.
SCTP =
∫
dVxdVx′
1
2
φASABφ
B. (2)
The CTP field equations in the presence of sources may be found by varying this
action with respect to the + and − fields. The fact that we can present a quasi-
particle prescription directly in terms of an action principle implies that it is a
well-defined canonical system, as one would expect from general renormalization
arguments. Following a similar line of argument to Lawrie [7], a general Gaussian,
quadratic form for a closed time path action may be expressed in terms of general
sources. Calzetta and Hu approached the problem in essentially the same fashion
using a more general formalism than Lawrie in [6]. These sources can be thought of as
external quantities or as the renormalized shadows of higher loop contributions due
to self-interactions. They must be non-local quantities in general. This is testified
to by the importance of hard-thermal loop effective actions which contain non-local
regulatory terms. The non-locality leads to power-law type dissipation [8], rather
than exponential dissipation, amongst other things so it is important to retain it
[1,8]. This seems to defy the usual wisdom of effective field theory, in which only
local terms are retained [9]. In fact there is no contradiction when care is taken in
choosing the order of the approximation to which one is calculating. One reserves
the ability to make an approximate local expansion later once the dynamical effects
of the nonequilibrium development have been better understood. We write
SAB(x, x
′) =
(
α̂ β̂
−β̂∗ −α̂∗
)
, (3)
where the indices A,B run over the ± labels of the CTP fields,
α̂ = (− +m2)δ(x, x′) + I(x, x′), β̂ = J(x, x′) +Kµ(x, x′)
↔
DK
µ (4)
and a Hermitian derivative has been defined to commute with the functionK µ(x, x′):
x
DK
µ ≡
x
∂µ +
1
2
x
∂µ Kν(x, x
′).
Kν(x, x′)
(5)
for time reversibility. Notice the general form of the ‘connection’ term in this deriva-
tive. This inhomogeneous, conformal structure crops up repeatedly in non-equilib-
rium development. The currents associated with sources I, J,K µ are not necessarily
conserved since their behaviour is not completely specified by the action, but the
action is differentially reversible. The sum of rows and columns in this operator
37
M.Burgess
is zero, as required for unitarity and subsequent causality. The significance of the
off-diagonal terms involving K µ can be seen by writing out the coupling fully:
Kµ(x, x′) ·
(
φ+D
K
µ φ− − φ−D
K
µ φ+
)
. (6)
The term in parentheses has the form of a current between components φ+ (the
forward moving field) and φ− (the backward moving field). When these two are
in equilibrium there will be no dissipation to the external reservoir and these off-
diagonal terms will vanish. This indicates that these off-diagonal components (which
are related to off-diagonal density matrix elements, as noted earlier) can be under-
stood as the mediators of a detailed balance condition for the field. When the term
is non-vanishing, it represents a current flowing in one particular direction, pointing
out the arrow of time for either positive or negative frequencies. The current is a
‘canonical current’ and is related to the fundamental commutator for the scalar field
in the limit + → −.
The system can be analyzed, and in principle solved, by looking for the Green
functions associated with this system. These can all be expressed in terms of the
Wightman functions G(±)(x, x′) using the relations
G(+)(x, x′) =
[
G(−)(x, x′)
]∗
= −G(−)(x′, x). (7)
The Wightman functions are the sum of all positive or negative energy solutions,
satisfying the closed time path field equations, found by varying the action above.
Thus they are the embodiment of the dispersion relation between k and ω = k0.
G̃(x, x′) = G(+)(x, x′) +G(−)(x, x′), G(x, x′) = G(+)(x, x′)−G(−)(x, x′). (8)
G(x, x′) is the sum of all solutions to the free field equations and, in quantum field
theory, becomes the so-called anti-commutator function. The symmetric and anti-
symmetric combinations satisfy the identities
x′
∂t G(x, x′)
∣
∣
∣
t=t′
= 0, (9)
and
x′
∂t G̃(x, x′)
∣
∣
∣
t=t′
= δ(x,x′). (10)
The latter is the classical dynamical equivalent of the fundamental commutation re-
lations in the quantum theory of fields. Other Green functions may be constructed
from these to model the processes of emission, absorption and fluctuation respec-
tively:
Gr(x, x
′) = −θ(t, t′)G̃(x, x′),
Ga(x, x
′) = θ(t′, t)G̃(x, x′),
GF (x, x
′) = −θ(t, t′)G(+)(x, x′) + θ(t′, t)G(−)(x, x′). (11)
It may be verified that, since G(+)(k, x) depends only on the average coordinate, the
commutation relations are preserved (see equation (10)) even with a time-dependent
38
Nonequilibrium quasiparticles
action. The general solution for the positive frequency Wightman function may be
written
G(+)(x, x′) = −2πi
∫
dn−1k
(2π)n−1
eikµx̃
µ (1 + f(k0, x))
2|ω| , (12)
where f(k0, x) is an unspecified function of its arguments and it is understood that
k0 = |ω| (this describes the dispersion of the plane wave basis). The ratio of Wight-
man functions describes the ratio of emission and absorption of a coupled reservoir
(the sources). As observed by Schwinger [10,11], all fluctuations may be thought of
as arising from generalized sources via the Green functions of the system. Thus a
source theory is an effective description of an arbitrary statistical system.
In an isolated system in thermal equilibrium, we expect the number of fluc-
tuations excited from the heat bath to be distributed according to a Boltzmann
probability factor [12].
Emission
Absorption
=
−G(+)(ω)
G(−)(ω)
= eh̄β|ω|. (13)
h̄ω is the energy of the mode with frequency ω. This tells us the rate flow from the
‘heat-bath’ in equation (6) is in balance. It is a classical understanding of the well-
known Kubo-Martin-Schwinger relation [13,10] from quantum field theory. In the
usual derivation, one makes use of the quantum mechanical time-evolution operator
and the cyclic property of the trace to derive this relation for a thermal equilibrium.
The argument given here is identical to Einstein’s argument for stimulated and
spontaneous emission in a statistical two state system, and the derivation of the
well-known A and B coefficients. It can be interpreted as the relative occupation
numbers of particles with energy h̄ω. This is a first hint that there might be a
connection between heat-bath physics and the two level system.
Finally, it is useful to define quantities of the form
Fµ =
∂µf
f
=
1
2
∂µ ln(f), (14)
Ωµ =
∂µω
ω
=
1
2
∂µ ln |ω(x)|. (15)
which occur repeatedly in the field equations and dispersion relations for the system
and characterize the average rate of development of the system. Note the similarity
in form to the connection term in the derivative of equation (5).
3. Inhomogeneous scaling and gauge formulation
Inhomogeneous field theory can be presented in a natural form by introducing
a ‘covariant derivative’ Dµ which commutes with the average development of the
field. This derivative is thus physical, in the sense of being a Hermitian operator.
This description parallels the structure of a gauge theory (in momentum space)
with a complex charge, generalized chemical potentials and quantum field theories
39
M.Burgess
in curved spacetime (see the local momentum space expansion approach of [6] and
the curved spacetime formulation of Nicola [14]). Consider the derivative
Dµ = ∂µ − aµ (16)
and its square
D2 = − ∂µaµ − 2aµ∂µ + aµaµ. (17)
Derivatives occur in the field equations and in the dispersion relation for the field
and they are thus central to the dynamics of the field and the response (Green)
functions. As with a gauge theory, the effect of derivatives on spacetime dependent
factors may be accounted for in a number of equivalent ways, by redefinitions of
the field. In a gauge theory, we call this a gauge transformation and we usually
demand that the theory be covariant, if not invariant under such transformations.
In a nonequilibrium field theory, we require only covariance, since it is normal to
deal with partial systems in which conserved currents are not completely visible and
thus invariance need not be manifest.
In order to solve the closed time path field equations, it is useful to solve the dis-
persion relation, giving the physical spectrum of quasi-particles in the system. In the
Keldysh diagrammatic expansion of Schwinger’s closed time path generating func-
tional, one expands around free particle solutions. By starting with a quasi-particle
basis here we can immediately take advantage of resummations and renormalizations
which follow from the unitary structure (specifically two-particle irreducible or daisy
diagrams). It also allows one to track changes in the statistical distribution through
the complex dispersion relation, instead of using real Vlasov equations coupled to
real equations of motion.
Let us briefly review the formalism introduced in [1]. In order to make its meaning
clearer we can simplify the notation and gloss over some of the details which are not
essential to the conceptual makeup. For a technical discussion readers are referred
to [1], with the change of notation
(I, J,Kµ) → (A,B, γµ). (18)
Substituting the form of the Green function in equation (12) into the equation of
motion, one obtains the dispersion relation
k2 +m2 + I(k, x) +
i
2
(∂µI)(T
µ − vµg /ω)− iJ̃ − ∂µKµ
− (F − Ω)2 − 2ikµ(F − Ω)µ − 2ikµKµ −K
µ
(F − Ω)µ = 0, (19)
where the quantities within are defined by
Tµ =
∂f/∂kµ
(1 + f)
, vµg =
∂ω
∂kµ
= ki/ω(k), T µ − vµg /ω =
1
G(+)(k)
∂G(+)
∂kµ
. (20)
Notice that the gradient of the plasma distribution or spectral envelope T µ is re-
sponsible for classical Landau damping of the field modes due to scattering from the
40
Nonequilibrium quasiparticles
wave modes of I(x, x′). Also, the imaginary part of this dispersion relation is just
the generalization of the Vlasov transport equation
kµ∂µ f(k, x) = (1 + f)kµFµ = ω C, (21)
[∂t + vig∂i] f = C, (22)
where C is a ‘collision term’ (see, for instance, [15]). The ‘collision terms’ are non-
zero here because the sources play the role of mediating interactions, just as Landau
damping is essentially collision by scattering off a third-party field. One could sep-
arate out the imaginary part and solve it as a separate equation but, rather than
do this, it is useful to retain a complex dispersion relation and complex ω(k). This
will make it possible to reveal which of the key players of this formulation have
equivalent roles in their effect of the dynamics below.
Without any approximation, it is straightforward to show that, in the general
inhomogeneous case,
(− +m2)G(+)(x, x′) = −2πi
∫
dn−1k
(2π)n−1
(1 + f)
2|ω| eik(x−x′),
[
− (ikµ + Fµ − Ωµ)
2 − ∂µ(ikµ + Fµ − Ωµ)
]
= 0. (23)
It is then natural to rewrite this by making the identification
aµ = Fµ − Ωµ +Kµ = −∂µSE(k) +Kµ . (24)
Indeed, it does not require a great leap of insight to see that all of these effects
come from the x-dependence of G(+) in equation (12). We can even extend the
generalization to include the case of a system of finite size which is expanding or
contracting, by letting
aµ = Fµ −Nµ − Ωµ +Kµ . (25)
The Nµ term then arises from the x dependence of the momentum space measure:
∫ dnk
(2π)n
→
∏
µ
1
Lµ
∑
lµ
(26)
giving a contribution
Nµ =
∂µ(L0 . . . Ln−1)
(L0 . . . Ln−1)
. (27)
This contribution makes a more intimate contact with quantum cosmological mod-
els of expanding universes and serves to emphasize the fact that scaling is a general
phenomenon, regardless of whether it is a homogeneous (global) rescaling or an
inhomogeneous (local) rescaling. The connection now embodies the effect of chang-
ing statistical distributions and quasi-particle energies (ω is solved in terms of the
sources through the dispersion relation) and the rarefaction of the field by expan-
sion through Nµ. It also shows that the source Kµ plays essentially the same role as
41
M.Burgess
these effects and it is thus capable of absorbing or “resumming” them. Furthermore,
the field aµ is related to a measure of the rate of increase of the entropy SE of the
mode. This in turn means that all of these are related to the behaviour of a density
matrix, since off-diagonal elements labelled by K µ represent a density matrix in this
formulation.
In terms of the covariant derivative, one now has:
(−D2 +m2)G(+)(x, x′) =
− 2πi
∫
dn−1k
(2π)n−1
(1 + f)
2|ω| {−(ikµ −Kµ)
2)− ∂µ(ikµ −Kµ)}, (28)
[
−
x
+m2
]
G(+)(x, x′)
+
∫
dVx′′
[
I(x, x′′)− J̃(x, x′′)−K
µ
(x, x′′)
x′′
∂µ −∂µKµ
]
G(+)(x′′, x′) = 0. (29)
In the inhomogeneous case there is no dispersion relation consisting of continuous
frequencies in general, so the dispersion relation will only exist for a discrete set.
It is convenient to divide the discussion into two parts: determining the dispersion
relation and the nature of the restricted set of values which satisfy the dispersion
relation.
The problem to be addressed is contained in the following form in momentum
space:
(− +m2)G(+)(x, x′) +
∫
dVx′′
dnk
(2π)n
dnp
(2π)n
eik(x−x′′)+ip(x′′−x′)
×S(p, x′′ + x′)G(+)(k, x+ x′) = λG(+)(x, x′).(30)
The integral over x′′ is no longer a known quantity in general, but it is possible
to extract an overall Fourier transform by shifting the momentum p → p + k and
defining the average variable of interest x = 1
2
(x+ x′):
(k2 + ikµ∂µ −
1
4
+m2)G(+)(k, x)
+
∫
dVx′′
dnp
(2π)n
ei(x
′′−x′)S(k, x+ x′′)G(+)(k + p, x′′ + x′) = λG(+)(k, x). (31)
In order to find eigenvalues, required for a stable expansion in Fourier space, it
is necessary to extract the factor of G(+)(k, x) from this expression. This is not
necessarily possible for arbitrary values of k. It is possible, however, if the momenta
are restricted to a set expressed by the property
G(+)(k + p, x′′ + x′) = G(+)(k, x′′ + x′). (32)
The differential equation satisfied by G(+)(x, x′) is thus, in terms of the new
derivatives,
[
−D2 +m2 +K
2
(k, x) + I(k, x)− J̃(k) +
i
2
(∂µI)(T
µ − vµg /ω)
]
k
G(+)(x, x′) = 0,
(33)
42
Nonequilibrium quasiparticles
where the appearance of the subscript k to the bracket serves to remind that the
equation exists under the momentum integral. This relation relates the time and
space dependence (lack of translational invariance) to the spectral content of the
field, i.e. it forges the link between x dependence and kµ dependence.
The presence of the D2 implies the existence of quasi-particles: an effective sym-
bol which replaces a conventional one. The positive frequency field may be ‘gauge
transformed’ using the integrating factor (Wilson line)
φ(k) → φ(k)e
∫
aµdxµ
. (34)
We might also refer to this as a quasi-particle transformation. It is also related to
renormalizations [16]. This shows the explicit decay (amplification) of the k-th field
mode. This transformation also has a nice physical interpretation in terms of the
entropy of the modes, defined above. The Wilson line is the negative exponential of
an entropy, showing how the field decays as the energy of a mode becomes unavailable
for doing work, i.e. as its entropy rises. Note however that this does not describe the
total entropy of the system, only a measure of the mode in question.
The above transformation should not be confused with similarity transformations
on the closed time path action. Because of unitarity, the plus and minus components
of closed time path fields have to satisfy a global O(1, 1) symmetry, which allows a
certain freedom in the way one chooses to set up the solution of the system. One can
choose, for instance, to work with Feynman Green functions and Wightman func-
tions, or with advanced and retarded functions, or with general mixtures of these.
The only constraint imposed by unitarity is that the sum of rows and columns in the
action (i.e. in the argument of the exponential in the generating functional) remains
zero when plus and minus labels are removed. The transformations considered here
are field redefinitions. This need not even be a symmetry of the system, since a
nonequilibrium system is often an incomplete (open) system.
The reason for the similarity in form between a gauge theory and a theory of
field rescalings is that both are linked through the conformal group. This also ex-
plains the connection to the curved spacetime approach already referred to: general
spacetime metrics are not of interest, but conformal rescalings are. Gauge theories
bear the structure of the conformal group, not the Lorentz group and time depen-
dent perturbations and changes of variable are also connected with inhomogeneous
rescalings of the conformal group. One will not normally see a conformal symmetry
in the original action because the effective field theories we are discussing are in-
complete: they describe partial systems, in which we ignore heat baths and external
influences etc. It can be argued that there one should study generalized descriptions
of nonequilibrium field theory such that they are fully invariant with respect to
conformal transformations. This should be explored further.
4. Real gauge theories
In order to further understand this source formulation and the meaning of the
effective quantities we have introduced, it is useful to look at a real gauge theory.
43
M.Burgess
The familiar example of scalar electrodynamics is chosen to preserve the scalar field
structures which have already been introduced, as well as to extend them to a simple
gauge symmetry. The action for scalar electrodynamics, in terms of a complex scalar
field Φ is,
S =
∫
dVt
{
h̄2c2(DµΦ)†(DµΦ) +m2Φ†Φ+
λ
3!
(Φ†Φ)2 + . . .+
1
4
F µνFµν
}
, (35)
where Dµ = ∂µ + ieAµ. This can also be written in terms of a two component field
φA:
S =
∫
dVt
{
1
2
h̄2c2(∂µφA)(∂µφA)− eh̄c(∂µφA)ǫABAµφB
+ e2φAφAA
µAµ +
λ
4!
(φAφA)
2 + . . .+
1
4
F µνFµν
}
. (36)
The field equations are easily worked out and supplemented with generalized sources,
just as in the scalar case. One thing which makes a true gauge theory different from
the scalar example is that it cannot be purely quadratic. The interaction between
scalar and vector field introduces third and fourth order terms. To simply drop these
would turn the example into a number of decoupled scalar theories, at least at the
pure state level. However, we may focus on the quadratic part of the model in an
open statistical ensemble, in which non-zero mean-fields are generated by virtue of
an external influence such as a heat bath, or ambient electromagnetic field. This
preserves some of the gauge structure even at the level of quadratic dynamical
fields, by introducing background mean-fields φa(x) and Aµ(x). These fields are not
necessarily equilibrium states.
An immediate comparision which one can make between electrodynamics and
the quasi-particle formalism, is between the current expressed in equation (6) and
the conserved current of electrodynamics. The J ·A coupling is reflected in the forms
J µ
11Aµ = eǫab(φa∂µφb)A
µ(x), J µ
12Kµ = ǫAB(φAD
K
µ φB)K
µ(x, x′). (37)
From this one sees that Kµ plays the part of a non-local current which operates
in the A,B space (emission/absorption). It trades off non-conservation in one part
of the system against non-conservation in another part and therefore provides a
mechanism for redistributing stuff (energy, charge etc.) around the system. This is
also closely analogous to the phenomenon of Landau damping. The electrodynamical
current can also be interpreted in the same way, but there it is a redistribution of
the phase of the field with respect to the U(1) symmetry which takes place. Owing
to the relationship between gauge symmetry and unitarity, this amounts to saying
that Aµ and Kµ are like each others’ imaginary complements.
To investigate the formal similarities further, it is useful to introduce a notation
to distinguish the indices. There are now two sets, the O(1, 1) indices A,B = +,−
which represent the closed time path unitarity space, and the U(1) electromagnetic
44
Nonequilibrium quasiparticles
indices a, b = 1, 2 which represent the gauge symmetry. They can be combined:
φA = (φ+, φ−), (38)
φa = (φ1, φ2), (39)
φa
A = (φ1
+, φ
2
+;φ
1
−, φ
2
−). (40)
We also have to deal with two flavours of field, the scalar and the vector field. These
can be grouped into a generic vector by defining:
Ψα = (φa, Aµ). (41)
The action then takes the much simplified form
SCTP =
∫
Ψα Sαβ Ψβ =
∫
Ψα
[
Sab Saν
Sµb Sµν
]
Ψβ. (42)
Each term is a two-by-two matrix with closed time path components. The two-by-
two structure is labelled by capital roman indices, i.e. the whole thing has indices
SAB. Introducing corresponding sources,
scalar → vector, I → Iµν , Kλ → Kλ
µν , (43)
it is possible to cast all of the fields in the same mould. The meaning of such
sources can be then examined in the context of scalar electrodynamics. If we consider
the terms which would appear in the different blocks for scalar electrodynamics
expanded around a background field φa and Aµ, the terms then look like equation (3)
with a generalized form:
Sab =
(
α̂ab β̂ab
−β̂∗
ab −α̂∗
ab
)
, Sµν =
(
α̂µν β̂µν
−β̂∗
µν −α̂∗
µν
)
, Saν =
(
α̂aν β̂aν
−β̂∗
aν −α̂∗
aν
)
,
(44)
where the terms appearing are schematically of the form,
α̂ab = (− +m2)δ(x, x′)δab + Iab(x, x
′) + eǫabδ(x, x
′)A
µ
(x)
↔
∂µ ,
β̂ab = Jab(x, x
′) +Kµ
ab(x, x
′)
↔
DK
µ +eǫabδ(x, x
′)A
µ
∂µ ,
α̂µν = − δ(x, x′)gµν + Iµν(x, x
′) + e2φ
2
(±) gµν ,
β̂µν = e2φ
2
(±) gµν + Jµν(x, x
′) +Kλ
µν(x, x
′)
↔
DK
λ ,
α̂aν = ǫabφ
(±)
b ∂µ ,
β̂aν = ǫabφ
(±)
b ∂µ . (45)
Generic sources I, J,K have, once again, been introduced. This is not to imply that
all of these sources will be required or relevant for every calculation. We include
them all here for generality, to show how they relate to more familiar terms, and
how the parts of linear combinations provide essentially equivalent contributions
45
M.Burgess
to the dynamics. To write out every permutation of A,B indices would only be
nebulous, so these permutations have been written (±) etc. It is difficult not to
confuse the various indices in these expressions, but the fact that this is so, also
indicates just how analogous their roles are. The point of this ‘gauge like’ formulation
is to acknowledge the covariance of underlying group transformations at work.
The derivative for the gauge field could have been written more symmetrically
using a notation like
↔
DA
µ , but for gauge theories there is always a gauge condition
of the form
∂µA
µ = χ(φa), (46)
which is required in order to give the vector Aµ representation of the Lorentz group
the same properties as the Fµν representation. This condition allows us to rewrite
connection terms in the form of other generic terms, already included, so there is no
harm in writing partial derivatives here. Comparing the terms in equation (45) it is
easy to see how the gauge field appear in much the same way as the sources Kµ. ǫab
is equivalent to a factor of i =
√
−1 in a complex formulation of the scalar field.
Since some interactions can be replaced by generic sources at the quadratic level,
a complex problem is abstracted into a simpler one, which can be discussed schemat-
ically, unencumbered by irrelevant detail. In particular we see how the background
fields act as sources themselves and how the dynamical fields ‘communicate’ or inter-
act via channels represented by these sources. Schwinger’s source theory was clearly
influenced by Shannon’s information theory, in which data are transmitted from
source to sink. Schwinger’s insight was to realize that such generic communication
is at the heart of all interaction between field fluctuations. Sources turn closed sys-
tems into open systems by allowing them to communicate either by overlapping,
or by contact at a boundary. This is the heat-reservoir model of thermodynamics,
thinly disguised and generalized.
5. Conformal covariance
In the preceding section we have identified a ‘covariant derivative’ with a pseudo-
gauge field aµ, for inhomogeneous (i.e. nonequilibrium) systems. This connection is,
in fact, a familiar object in field theories of all kinds. It typically occurs together
with transformations of the conformal group, or transformations which are position
dependent. The conformal group contains the Poincaré group as a subgroup, and its
importance is already known in connection with covariance of gauge theories, though
it is perhaps not widely appreciated, as exemplified by the long-standing confusion
over definitions of the energy-momentum tensor [17,18]. (Such confusion disappears
when one acknowledges the fact that the gauge theory really belongs to the confor-
mal group, not the Poincaré group, because of the freedom to perform spacetime
dependent gauge transformations at arbitrary points.) Conformal transformations
are usually expressed in terms of a position dependent scaling of the metric tensor:
gµν(x) → Ω2 gµν(x). (47)
46
Nonequilibrium quasiparticles
Most discussions of conformal symmetry are restricted to two dimensional Euclidean
space where the conformal symmetry is automatic for analytic functions by virtue
of the Cauchy-Riemann relations. This restriction is not relevant here since we do
not specifically need to use the analytic properties of the fields. The inhomogeneous
(Abelian) part of a gauge transformation U has the form
Φ → UΦ,
Aµ → Aµ + U−1(∂µU); (48)
in general relativity the trace of connection is
Γσ
λσ =
1
2
g−1
µν ∂λ gµν , (49)
giving a transformation rule of
Γσ
λσ → Γσ
λσ +
∂λΩ
Ω
gµµ. (50)
Note also the form of equation (20) which relates to Landau damping, which is
similar in the conjugate space. Clearly the form,
Γµ = ∂µ lnX =
∂µX
X
. (51)
has a general significance. Indeed, logarithms which add scaling corrections are fa-
miliar in a different context: the renormalization group. There they are picked up
by analytic continuation away from spacetime dimensionalities where correlation
‘loops’ are scale invariant. This can be seen by power counting. In a nonequilibrium
theory, what is perhaps surprising is that these renormalization group logarithms,
and the spatial inhomogeneity logarithms seen earlier, are connected. While k and x̃
are conjugate variables, k and x are a complementary pair, related by the dispersion
relation.
The main reason for pointing out this connection is to motivate a deeper study
of conformally covariant theories, but practical insights might also be made pos-
sible. In [19] it was shown, in an explicit example, how an inhomogeneous scaling
transformation can be used to provide an exact quasi-field solution to the two-level
atom, without having to adopt the standard rotating wave-approximation.
6. Beyond quasi-particles
Can a quasi-particle field theory be shown to have a significance beyond elemen-
tary and composite excitations of fields? Could we model other dynamical systems
with more complex interactions, understanding how the nature of the dynamical
entities changes with the nonequilibrium development?
One current area of interest is in biology. In biological development, very small
interactions within cells induce changes at the molecular level (genes/proteins) whose
47
M.Burgess
final consequences lead to large and very dramatic changes in the way a total system
develops macroscopically (phenotype). What starts as a few proteins bumping into
one another, ends up as plant and animal life. The transition is a dramatic one and
clearly cannot be described by any simple field theory because it involves complex
interactions with time varying boundary conditions coupling a whole hierarchy of
scales. In equilibrium one could coarse grain cells into a ‘flesh-field’ and things
would then be well-described by a continuum hypothesis, but during development
(far from equilibrium) that cannot be true. No simple, local field theory gives rise
to such complicated, reproducible self-organization from a microscopic code.
What occurs in the evolution and development of biological organisms must
therefore be due to the complicated time-dependent interactions between neighbour-
ing composite objects, whose boundaries communicate modifications to dynamics for
the duration of their contact. While each arbitrary part of a system follows micro-
scopic laws in every detail, the totality of a complex system exceeds the sum of
its parts because it involves structural information about how to put those parts
together, i.e. the boundary conditions between neighbouring elements in a system.
Cooperation and competition between neighbouring cells introduces huge compli-
cations. Could any recognizable features of this process be reproduced in a field
theoretical model with time-dependent boundary conditions? This is clearly a tall
order, but with small progressive steps it might be possible to fill in some of the
‘magic’ behind biological development, which is currently beyond field theoretical
models.
The wisdom of effective field theory is that dynamical behaviour is, for many
purposes, independent of the underlying nature of its key players. Precise details of
interactions can be ignored to a determinable level of approximation. Effective field
theories are particularly applicable to statistical systems where coarse graining is a
central feature. Sources provide us with a way of modelling external influences. Could
they be further used to model cells which interact through permeable cell bound-
aries? Cells, after all, have the generic structure of quasi-particles: a core surrounded
by a cloud of screening material. Could a dynamical model, with appropriate inter-
actions, provide a schematic description of biological systems by modelling them
internally and externally, i.e. from the viewpoints of a cell and of a cluster of cells:
• A super-system of time-dependent cells which interact on contact.
• A single cell which experiences time-dependent interactions on its boundary
due to other cells.
Such constructions might be useful in modelling the immune system, for instance.
Whether or not this approach can lead to usable biological models is one thing, but
in any event it would be a useful testing ground for nonequilibrium analyses at more
microscopic levels, where field theory is prevalent. It is also important, in principle,
to understand how the microscopic to macroscopic transition occurs in detail in
genetic development.
48
Nonequilibrium quasiparticles
7. Conclusions
A proper study of field theories which are covariant with respect to a conformal
symmetry is needed, in order to understand nonequilibrium field theory. To date no
specific detailed studies have been carried out in more than two dimensions.
A natural place to begin would be to take the simplest case of the quasi-gauge
formalism presented here. This would be a quadratic scalar field theory with some
specified time-dependent, external influence which can be thought of as a “boundary
condition”. This problem has already been discussed in [19–22] in the context of
quantum optics, but there has not been sufficient occasion to take these studies to
an extensive conclusion. The simplest cases are those where one has a predetermined
model of what the sources should look like, based on prior knowledge of the external
environment to which a field theory is coupled. Hopefully more instructive examples
can be found so that one might explore the statistical behaviour of a system in
response to time-dependent interactions with external agent, such as in a biological
cell interaction. Again it is expected that a covariant formulation would lead to
insights about which effects are relevant and which are merely detail.
It is, of course, natural to suppose that changes of variable, i.e. changes of per-
spective will lead to transformations analogous to gauge transformations. After all,
we are perturbing a system with sources which vary in space and time. This is
precisely the nature of a gauge theory. Methods of solution which rely on diagonal-
ization of the action will also involve transformations which depend on space and
time. All such transformations demand covariant derivatives and transforming aux-
iliary fields. The amplification of modes makes this a recipe for a kind of space-time
dependent renormalization group. Some authors have suggested making coupling
constants run with time, but there are canonical restrictions associated with making
coupling constants depend on space-time coordinates [16]. The idea of completion
by general covariance is also reminiscent of the Vilkovisky-DeWitt effective action
[23]; probably this also has an interpretation in nonequilibrium physics.
All we have presented here is formalism, but formalism with an explanatory value
is not to be sniffed at. Whether practical benefits might emerge from quasi-particle
descriptions in nonequilibrium field theory is an open question. More work is needed
to overcome both the technical and conceptual difficulties involved.
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Нерівноважний принцип дії для квазіполів
М.Бурґес
Коледж м. Осло, Корт Аделерс ґейт, 30, 0254 Осло, Норвегія
Отримано 1 червня 1999 р.
Квазічастинкові описи нерівноважної фізики дозволяють простежу-
вати відповідні ступені вільности в обчисленнях статистичних систем
і забезпечують важливий формальний підхід до квантової теорії поля
поза рівновагою. Тут запропоновано формулювання нерівноважної
квазічастинкової теорії поля. Для розвитку схематичних теорій поля,
здатних описувати реальні системи, використовуються швінґерівсь-
кий твірний функціонал за шляхами, замкненими в часі, і узагальнені
джерела. Обґрунтовується важливість методу джерел для ефектив-
них нерівноважних теорій, а також його застосовність до біологічної
динаміки.
Ключові слова: нерівноважна теорія поля
PACS: 03.70.+k, 05.30.-d, 05.70.Ln
50
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