Ann. Henri Poincaré 4, Suppl. 2 (2003) S863 – S880
c Birkhäuser Verlag, Basel, 2003
1424-0637/03/02S863-18
DOI 10.1007/s00023-003-0967-1
Annales Henri Poincaré
Thermal Aspects in Quantum Field Theory
Jacques Bros∗
Abstract. Thermal (or “KMS”) states as well as ground states are characterized
by analyticity properties in the (complexified) time variable. Such a characterization is applied to the quantum field theoretical systems on Minkowski, de Sitter
and anti-de Sitter spacetimes. Privileged theories (or “vacua”) can be defined on
the basis of general principles which ensure “maximal” analyticity properties of
the correlation functions. In such theories, there exists an observer-dependent thermal interpretation of the “vacuum” which is due to the (complex) geometry. In
Minkowski spacetime, the (non-privileged) thermal quantum field theories at arbitrary temperature are investigated for their particle aspect at asymptotic times.
This aspect is encoded in the corresponding two-point functions through a certain
“damping factor”, which is shown to depend on the dynamics of the interacting
fields and suggests a possible substitute to the usual pole-particle concept in the
thermal case.
The first part of this talk (Sec 1–3) will be devoted to the following question: how
does one determine the properties of stability of quantum states for relativistic
systems which are described by local quantum fields, either in the Minkowskian
background of flat spacetime, or more generally in certain curved spacetimes of
simple type considered as given backgrounds for the quantum systems?. There is
a general result of Pusz and Woronowicz [8] on quantum systems, according to
which the class of quantum states satisfying an appropriate criterion of stability
(called “passivity”) can be partitioned into two subclasses, namely, on the one
hand the “ground states” and on the other hand the “thermal equilibrium states”
or “KMS states”; both cases are characterized by specific analyticity properties in
the time-variable of the correlation functions of all pairs of local observables of the
system. When one deals with a relativistic system in which the local observables
are described in terms of quantum fields, it turns out that the “ground-state” or
“thermal-state” interpretation of a stable state of the theory will in general depend on the motion of the local observer. While this phenomenon already occurs
in flat spacetime with the Unruh effect [9], it seems to acquire a more general
validity in curved spacetime by appearing as the manifestation of a certain type
of “temporal curvature” of the world lines which is felt as a thermal effect by the
corresponding observer. In models of spacetime equipped with a maximal symmetry group such as Minkowski, de Sitter and anti-de Sitter spacetimes, a specially
∗ The results presented in the first part of this talk have been obtained in joint works with
Henri Epstein, IHES- Bures-sur-Yvette (France), and Ugo Moschella, Università di Insubria, Como
(Italy) [1-4]. Those presented in the second part have been obtained in joint works with Detlev
Buchholz, Universität Göttingen (Germany)[5-7].
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favourable circumstance occurs. In fact, one can then define classes of privileged
theories (Sec 3) whose stability properties are encoded in appropriate global analyticity properties of the correlation functions with respect to the complexified
spacetime variables. In particular, the two-point functions of the fields have “maximal analyticity properties” which can be expressed through a Källèn-Lehmanntype integral representation. In such theories, there exists a stable state invariant
under the full symmetry group which may legitimately be called a “generalized
vacuum”, the word “vacuum” being used here in the mathematical sense of the
GNS-construction [10], as explained below. The global analyticity properties play
the role of a “generalized relativistic spectral condition”, although such a term may
be misleading: in fact, it is shown that for all these theories, there exists a complete
observer-dependent interpretation of the “vacuum” either as a ground state or as
a thermal equilibrium state (the former being never satisfied in the de Sitter case):
this interpretation rigorously results from the combination of analyticity with the
(complex) geometry of the spacetime manifold.
The second part of this talk (see Sec 4) deals with the description of particles
created by a quantum field in a thermal background in Minkowski spacetime. We
shall in particular propose an answer to the following question: what becomes of the
asymptotic particle aspect of quantum fields and of the characterization of particles
in terms of poles of the two-point functions, if one combines the basic principles
of relativistic quantum field theory with the general KMS condition expressed in
the Lorentz frame of the thermal bath? We shall summarize here the results of
[7], in which it is shown how the asymptotic particle aspect of the theory can be
determined in a specific way by the dynamics of the interacting fields through a
certain damping factor occurring in a Kallen-Lehmann-type representation of the
two-point function.
1 Analyticity in complex time as a criterion for stability:
ground states and KMS-states
Imaginary time formalism is often presented in the physical literature as being
dictated by considerations of convenience: convenience of the use of the Euclidean
metric in place of the Minkowskian metric of real space, convenience of the compactness of the integration interval [0, β] (β being the inverse temperature) in the
study of quantum statistical systems.
Here we first wish to emphasize that the analyticity properties of correlation
functions with respect to a complexified time variable are actually the necessary
manifestation of a basic principle of stability for the states of the quantum system
under consideration [8].
Let A be the algebra of all local observables A of a given quantum system.
We adopt the Heisenberg picture, in which the evolution of the system is described
by a group of “time automorphisms” αt acting on these observables: A → αt (A)
(α0 (A) = A). Each stable state ω generates a Hilbertian representation of the
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algebra of observables of the system, in which it appears as a set of expectation
values ω(A) ≡< A >ω satisfying the condition that ω(αt (A)) ≡< A(t) >ω is independent of t. Here we have identified the abstract algebraic notation in which
the state ω is seen as the action ω(A) of a positive linear functional on the algebra of observables, with the usual Hilbert space notation < A >ω in which A
(resp. A(t)) is meant as the operator representative of the observable A (resp.
αt (A)). The fact that any state ω (including the thermal equilibrium states) can
be represented as a distinguished vector >ω of a certain Hilbert space Hω , which
is spanned by the action of the whole algebra of observables on that particular
vector corresponds to a standard mathematical construction, called the GelfandNaimark-Segal (GNS) construction (see e.g. [10] and references therein); the vector
>ω is then called a “GNS-vacuum” (see also the remark below). In the Hilbert
space Hω , the automorphisms αt are represented by unitary operators Uω (t) such
that A(t) = Uω (t)AUω (t)−1 .
We stress the conceptual importance of the “passivity” criterion of stable
states introduced in [8] in terms of the reaction of the system in these states to external perturbations. According to [8] this criterion then allows one to distinguish
two types of stable states ω, namely the “ground states” which satisfy a condition
of energy boundedness from below or spectral condition, and the “thermal equilibrium states” which satisfy the so-called KMS-condition [11]. It is remarkable that
these two types of states are characterized by specific analyticity properties with
respect to the time variable t of all the two-point correlation functions of the form
.
.
(t) = ω(Bαt (A)) ≡< B A(t) >ω ,
WAB (t) = ω(αt (A)B) ≡< A(t) B >ω and WAB
(A, B) denoting any pair of local observables of the system.
±
(t),
In both cases, there exists for each pair (A, B) two analytic functions WAB
defined respectively below and above the real axis in the complex t−plane, such
−
+
that for all real values of t, WAB (t) = lim WAB
(t− iη) and WAB
(t) = lim WAB
(t+
iη) for η tending to zero with positive values. However the following difference
holds:
−
+
(t) and WAB
(t) are respectively holomorphic in the
a) Ground states: WAB
half-planes mt < 0 and mt > 0.
−
+
(t) and WAB
(t) are
b) Thermal equilibrium states of temperature β −1 : WAB
−
respectively holomorphic in the strips Σβ = {t; −β < mt < 0} and Σ+
β =
{t; β > mt > 0}, and moreover the following KMS periodicity condition
+
−
(t + iβ) (implying also WAB
(t) = WAB
(t − iβ) and for
holds: WAB (t) = WAB
+
+
−
every t in Σβ : WAB (t) = WAB (t − iβ)).
.
(t) be
Let the commutator functions CAB (t) =< [A(t), B] >ω = WAB (t) − WAB
introduced (for each pair (A, B)). Then in the Fourier conjugate variable of t,
namely the energy variable E, the previous properties can be equivalently ex
(E), C̃AB (E)
pressed as follows in terms of the Fourier transforms W̃AB (E), W̃AB
of WAB (t), WAB (t) and CAB (t):
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a) Ground states:
W̃AB (E) = θ(E) C̃AB (E),
W̃AB
(E) = −θ(−E) C̃AB (E);
b) Thermal equilibrium states of temperature β −1 :
W̃AB (E) =
1
C̃AB (E),
1 − e−βE
W̃AB
(E) = −
1
C̃AB (E).
1 − eβE
The two cases a) and b) correspond to two different splitting procedures for
C̃AB (E), the former being a sharp support splitting (expressing energy positivity), while the latter is a smooth support splitting with exponential tails specified
by the Bose-Einstein factor [1 − e−βE ]−1 .
Remark The KMS condition of the case b) was originally derived for the case of
statistical systems in a box, by using the standard Hilbertian formalism in terms
of the Hamiltonian H and of the density matrix e−βH for representing the action
of the thermal state ω, namely
ωβ (A) =
Tre−βH ρA
.
Tre−βH
The validity of the same analytic structure for the general case of infinite systems
was then established in [11]: in the new GNS Hilbert space Hω associated with
ωβ , this state is no longer represented as a mixture (or density matrix) but as a
vector state, namely the GNS-vacuum >ω .
2 Stability and analyticity in time variables
for relativistic quantum fields
We adopt the viewpoint of a general field theory described in terms of one (or several) Wightman-type quantum field Φ(x). The basic local observables
of the theory
are then linear combinations of field monomials of the form dx1 . . . dxn Φ(x1 )
. . . Φ(xn ) f (x1 , . . . , xn ), such integrals being understood as the action of operatorvalued distributions on smooth test-functions f with compact support.
If the background spacetime is Minkowski spacetime and t denotes the time
coordinate along a time-axis with unit vector e, the general stability criterion
described in Sec.1 then leads us to consider all the pairs of correlation functions
or “Wightman functions” (understood in the sense of distributions) of the form
Wmn (t) =< Φ(x1 + te) . . . Φ(xm + te)Φ(xm+1 ) . . . Φ(xm+n ) >ω
and
Wmn
(t) =< Φ(xm+1 ) . . . Φ(xm+n )Φ(x1 + te) . . . Φ(xm + te) >ω ,
where the state >ω (characterized mathematically as a GNS-vacuum for the representation of the field algebra, as explained above) is stable, namely invariant
under the translations of time along the e−axis.
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The fact that this state is a ground state or a KMS-state is then specified by
the corresponding analyticity property with respect to the complexified variable t
of the previous pairs of Wightman functions, as described in Sec.1.
More generally, one can consider families [γ̂] of world-lines γ parametrized
by a proper-time parameter t and which are orbits of a certain (one-parameter)
isometry group Γ of the underlying spacetime X. Denoting by x = x(γ) (t) the
parametrization of each trajectory γ in [γ̂] and putting for simplicity Φγ (t) =
Φ(x(γ) (t)), one is led similarly to characterize the stability properties of a given
field theory on X with respect to the family of orbits [γ̂] of the “evolution group”
Γ by the analyticity properties of the pairs of Wightman functions
Wmn (t) =< Φγ1 (t) . . . Φγm (t)Φγm+1 (0) . . . Φγm+n (0) >ω
and
(t) =< Φγm+1 (0) . . . Φγm+n (0)Φγ1 (t) . . . Φγm (t) >ω
Wmn
in the corresponding complexified variable t.
We emphasize that the “ground state” (or “zero-temperature”) character or
the “thermal” character (at temperature T = β −1 ) of the state >ω , corresponding
respectively to the analyticity of Wmn (t) and Wmn
(t) in half-planes or in periodic
KMS-strips (with width β) of the t−plane, is now observer dependent, since relative to the family of world-lines [γ̂]. Such a family may indeed cover possibly only
a part of the spacetime X, which corresponds to the existence of an “horizon”
for the corresponding observers. For these observers, the corresponding energy interpretation of the state is done in the Fourier conjugate variable E of t, which
is relative to the “evolution group” Γ considered and generally does not have a
global meaning with respect to X.
3 Privileged Quantum field theories in Minkowski,
de Sitter and anti-de Sitter spacetimes
Of course, it is always possible to complexify the proper-time variable t of any given
world-line of any spacetime manifold X, and if families of world-lines associated
with an isometry group Γ of X exist, which is not always the case, one can introduce
the previous notions of stability relatively to Γ with arbitrary temperature (positive
or equal to zero): these notions rely on a partial complexification in one variable
of the manifold X.
Such complexification of time variables becomes particularly interesting if
the manifold X is analytic, namely if one can speak of global complexifications of
X in all the variables; in such a case, the complexification of the proper-time of a
world-line γ may define a corresponding “complex world-line” γ (c) as a curve in a
complex manifold X (c) which is a global complexification of X.
We shall consider three simple models of spacetime X which admit a global
complexification X (c) : Minkowski, de Sitter and anti-de Sitter spacetimes. They
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also have in common to admit a global symmetry group G, which extends to
a complex symmetry group G(c) of X (c) . For Minkowski, this is the (restricted)
Poincaré group acting on R4 . For de Sitter, which can be represented as the
one-sheeted hyperboloid with equation x2 ≡ x20 − x21 − x22 − x23 − x24 = −R2
embedded in the Minkowskian space R5 , the symmetry group G is SO0 (1, 4).
For anti-de Sitter, which can be represented by the quadric with equation x2 ≡
x20 − x21 − x22 − x23 + x24 = R2 embedded in R5 , the symmetry group G is the
pseudo-orthogonal group SO0 (2, 3). R can be called the radius of the corresponding de Sitter or anti-de Sitter spacetime. These three spacetimes admit classes of
privileged quantum field theories (QFT) , and corresponding GNS-vacua, whose
definition (given below) will benefit from the global symmetry group and from the
global complexified structure of these spacetimes. As a matter of fact, the correlation functions of such privileged QFT will satisfy properties of covariance under G
and analyticity domains which are the analogs of those expressing the relativistic
spectral condition in the Minkowski case.
Then for the spacetimes X which we consider, there are various “evolution
isometry groups” Γ, which all appear as subgroups of the global symmetry group G.
Each one-parameter group Γ admits a complexified group Γ(c) , which is a subgroup
of G(c) , and whose orbits will be complex world-lines γ (c) in the corresponding
manifold X (c) . The following typical situations will occur.
3.1
Two simple types of complexified orbits γ (c)
for the “evolution groups”
i) “straightlines” or “circles”: the complex orbits γ (c) are (topologically) isoCt
morphic to Ct or to 2πrZ
ii) “hyperbolae”: the complex orbits γ (c) are (topologically) isomorphic to
Ct
2πiρZ
(In the latter, t always denotes the proper-time variable)
The simplest example is provided by the complexified Minkowski spacetime
X (c) = C4 , in which one can distinguish:
(c)
a) All the orbits of (complexified) time-translation groups Γe , e being any unit
timelike vector in the forward cone V + , namely all straightlines of the type δ : z =
te + b (b being any real vector); they correspond to all possible uniform motions
in the kinematics of special relativity.
b) Classes of orbits of complexified one-parameter Lorentz subgroups, such as
(c)
the group Γ0,1 = SO0 (1, 1)(c) of pure Lorentz transformations in the coordi(c)
nates (z0 , z1 ); these orbits are the complexified timelike hyperbolae hρ : z0 =
ρ sinh ρt , z1 = ρ cosh ρt , with ρ, z2 , z3 constant and real. Each of these complex
curves contains two disjoint real world-lines, which are the corresponding two real
(c)
branches hρ , hρ of the complex hyperbola hρ . They are respectively represented
in the t−plane by the real t−axis (mod. 2πiρ) and the parallel to the real axis
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passing at the point πiρ but with negative orientation (mod. 2πiρ); hρ and hρ
are interpreted as world-lines of uniformly accelerated motions with acceleration
(c)
a = ρ1 . The complex curve hρ contains a “circle” z0 = iρ sin τρ , z1 = ρ cos τρ ,
(obtained for t = iτ and contained in the imaginary-time Euclidean spacetime
of X (c) ); this circle will introduce a 2πiρ−periodicity , which the corresponding
uniformly accelerated observer living on hρ (or on hρ ) may interpret in a thermal
a
way (namely as a KMS-type periodicity with the temperature T = 2π
) provided
its local field observables are in a GNS-vacuum state satisfying the corresponding
(c)
global analyticity property in the complex manifolds hρ . As seen below, this will
(c)
indeed be satisfied by all the privileged QFT’s on X and correspond to what is
known as the “Unruh effect”[9], but one must not confuse the geometrical period(c)
icity of the complex world lines hρ with the corresponding KMS property of the
QFT’s which will benefit from that periodicity.
In de Sitter spacetime the situation is as follows. All the isometry groups
Γ which admit families of time-like orbits interpreted either as inertial or uniformly accelerated motions are subgroups of pure Lorentz transformations in G =
SO0 (1, 4). All the corresponding orbits are world-lines of type ii), i.e. hyperbolae; for such a given group, they are obtained by taking all the sections of X
by a corresponding family of parallel (timelike) two-planes in the Minkowskian
space R5 , and only one of these sections, namely the “meridian hyperbola” in
the two-plane containing the origin, corresponds to a pair of inertial motions (or
geodesics). To be specific, it is sufficient to consider the complexified orbits of
(c)
the group Γ0,1 = SO0 (1, 1)(c) of pure Lorentz transformations in the coordinates
(c)
(z0 , z1 ) (similar to those of Minkowski space) of the form: hρ,e : z0 = ρ sinh ρt , z1 =
1
ρ cosh ρt , (z2 , z3 , z4 ) = (R2 − ρ2 ) 2 e, with e and ρ constant and real, e2 = 1 and
0 < ρ ≤ R. On the corresponding real world-lines hρ,e , hρ , e, the acceleration is
12
a = ρ12 − R12 .
In anti-de Sitter spacetime the situation is more diversified. The isometry
groups Γ which admit families of time-like orbits interpreted either as inertial or
uniformly accelerated motions are of three types, and the corresponding orbits
are sections of X by two-planes of R5 which present the three possible shapes of
conical sections.
a) All the subgroups which are conjugates with respect to SO0 (2, 3) of the rotation
group SO0 2 in the coordinates (z0 , z4 ) give rise to complexified orbits of type i),
Ct
isomorphic to 2πrZ
: the corresponding world-lines are in fact circles (or ellipses),
which are interpreted as uniformly accelerated motions with acceleration a such
that 0 ≤ a < R1 , the only geodesics being the “meridian circles” such as z02 + z42 =
R2 , z1 = z2 = z3 = 0. It is to be noted there that the replacement of the antide Sitter spacetime by its universal covering X̃ suppresses the “unphysical time-
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periodicity” of the previous circular world-lines: in fact, it replaces them by their
Ct
universal covering R, and correspondingly it replaces the complexified orbits 2πrZ
by Ct .
b) All the horocyclic subgroups which are conjugates with respect to SO0 (2, 3) of
the subgroup acting in the two-planes parallel to z0 +z1 = 0, z2 = z3 = 0 give rise to
complexified orbits of type i), isomorphic to Ct : the corresponding world-lines are
in fact parabolae, interpreted as uniformly accelerated motions with acceleration
equal to a = R1 .
c) All the subgroups which are conjugates with respect to SO0 (2, 3) of the Lorentz
subgroup G = SO0 (1, 1) in the coordinates (z0 , z1 ) give rise to complexified orbits
(c)
of type ii); for this typical subgroup, these are the hyperbolae of the form hρ,e :
1
z0 = ρ sinh ρt , z1 = ρ cosh ρt , (z2 , z3 , z4 ) = (ρ2 + R2 ) 2 e, with e and ρ constant and
real, e2 = e24 − e22 − e23 = 1 and ρ > 0. The corresponding world-lines are (as in the
Minkowski and de Sitter cases) branches of hyperbolae hρ,e , hρ,e , now interpreted
12
as uniformly accelerated motions with acceleration equal to a = ρ12 + R12 .
3.2
Definition of the privileged scalar quantum field theories
In Minkowski spacetime, the “privileged quantum field theories” are supposed to
satisfy the Wightman axioms[12] (or generalized versions of them, such as those of
the Jaffe fields[13] including the possibility of wilder short-distance singularities).
In de Sitter and anti-de Sitter spacetime, it is possible to introduce similar classes
of QFT’s [1,3,4], which allows us to give here a brief unified presentation for the
three spacetimes; we use the general notations X and X (c) for anyone of them.
We restrict ourselves for simplicity to the case of a single scalar field.
A) Postulates on the field Φ(x)
A basic postulate which is of general nature in QFT is the quantum formulation of Einstein causality, called local commutativity: the commutator [Φ(x1 ),
Φ(x2 )] is supposed to vanish (as an operator-valued distribution) in the region of
X × X in which x1 and x2 are spacelike separated. In all three cases, this region
can be defined by the condition (x1 − x2 )2 < 0; for the dS and AdS cases, the
latter is understood in terms of the quadratic form of the corresponding ambient
space R5 (the quadric X being defined by x2 = ±R2 in that space) .
For a scalar field, the covariance assumption can be written for all three cases
as: U (g)Φ(x)U (g)−1 = Φ(gx), where U (g) denotes a unitary representation of the
global symmetry group G in the Hilbert space of states.
Massive free fields satisfy the Klein-Gordon equation (DX + µ2 )Φ = 0, where
DX denotes the Laplace-Beltrami-type operator on the corresponding spacetime
X. For the dS and AdS cases, this is the trace on X of the G−invariant Laplacian
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(or D’Alembertian) in the corresponding ambient space R5 . Additional specifications on the range of the mass variable µ are given below.
B) Postulates on the vacuum state Ω
The vacuum state Ω is supposed to be invariant under the group G.
The n−point Wightman functions Wn (x1 , . . . , xn ) =< Ω, Φ(x1 ) . . . Φ(xn )Ω >
of the field Φ are supposed to satisfy specific global analyticity properties in the
complexified spacetime variables, which can be called generalized spectral condition. This condition requires that for n ≥ 2, each distribution Wn is the boundary
value of a holomorphic function Wn (z1 , . . . , zn ), defined in a certain “tuboid domain” Tn . We now give a precise definition of the latter for the three cases, together
with the corresponding interpretation
a) Minkowski spacetime M4 : We postulate the usual relativistic spectral condition,
which is equivalent[12] to the analyticity of the function Wn (z1 , . . . , zn ) in the tube
Tn (M4 ) defined by the conditions mzj+1 − mzj ∈ V + , 1 ≤ j ≤ n − 1.
b) de Sitter spacetime: The causal structure in X is inherited from that of the
ambient Minkowskian space R5 , namely the light-cone emerging from a point x in
X is the intersection of the Minkowskian light-cone emerging from x in R5 with
X. Correspondingly, one also shows that there exists complex “tuboid domains”
bordering the whole real spacetime X, which are of the form T + = T + ∩ X (c) and
T − = T − ∩ X (c) , where T + and T − are respectively the “forward” and “backward
tubes” in the complexified Minkowskian space C5 , defined by the condition mz ∈
V + or −mz ∈ V + . The following generalized spectral condition is then suggested
[1,3]: for each n, the function Wn (z1 , . . . , zn ) is analytic in a domain Tn , which is
the intersection by X (c) × · · · × X (c) of the five-dimensional Minkowskian tube
domain Tn (M5 ).
c) anti-de Sitter spacetime: A genuine spectral condition can be formulated in
terms of all isometry groups Γ of rotation type (listed under a) in the paragraph
of subsection 3-1 devoted to AdS), whose complex orbits can be seen to generate a
union of tuboid domains T + and T − in X (c) , defined by the condition (mz)2 > 0.
The corresponding formulation of the generalized spectral condition in terms of
analyticity domains of the functions Wn (z1 , . . . , zn ) is then also feasible [4], the
domains Tn of X (c) × · · · × X (c) being then defined in terms of the tuboids T +
and T − .
3.3
Two-point functions and thermal properties
of the privileged field theories
By exploiting the generalized spectral condition together with the covariance of
the fields and of the vacuum state, one can show that the two-point functions
W2 (z1 , z2 ) of all privileged QFT’s enjoy a maximal analytic structure. What is
meant here is that the following properties, known as old basic results for the
Minkowski case [12] are also satisfied for the dS and AdS cases [1–4].
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i) Analyticity in a maximal cut-domain D of X (c) × X (c) , which is invariant under
G(c) ; D is the set of all pairs (z1 , z2 ) such that the G(c) −invariant variable ζ =
(z1 − z2 )2 belongs to the cut-plane Π = C \ R+ . In all cases, the restriction of the
cut ζ ≥ 0 to the reals is the set of all pairs (x1 , x2 ) which are timelike-separated on
X. Concerning the AdS case, this result actually holds for the two-point functions
of QFT’s on the covering of AdS, while for those on the “true” AdS itself, the cut
in the ζ− plane reduces to the interval 0 ≤ ζ ≤ 4R2 . In all cases, each two-point
function W2 (z1 , z2 ) of a scalar field is thus represented by a function of a single
complex variable w(ζ) holomorphic in the cut-plane Π.
ii) Privileged free fields: The two-point functions of free fields (with relevant mass
ranges) are completely determined so as to satisfy the previous maximal analyticity
property. They are shown to be proportional to Legendre-type functions Pλ or Qλ
of the (G(c) −invariant) scalar product zR1 · zR2 (linearly related to ζ). In the latter,
the subscript λ determines the squared mass µ2 of the corresponding free field.
While the first-kind functions Pλ are obtained for the dS case, the second-kind
functions Qλ are obtained for the covering of AdS, but only integral values of λ
occur for the “true” AdS (which corresponds to the above peculiarity of the cut
in ζ and is closely related to the real-time periodicity of such theories).
iii) Källèn-Lehmann-type decompositions for general privileged QFT‘s. The possibility of representing the two-point functions of general (interacting) privileged
QFT’s on X as linear superpositions over a certain mass range (with a weight
which is a positive measure) of free-field two-point functions necesitates a more
refined analysis. For the dS case, one can distinguish two disjoint bases of free
fields (i.e. of Legendre functions Pλ ), whose sets of masses correspond respectively
to the labelling of the principal series and of the complementary series of unitary
irreducible representations of G. Fields whose two-point functions can be expanded
on these two disjoint bases differ from each other by their asymptotic properties at
large times on de Sitter spacetime. For the AdS case, there exists correspondingly
a general decomposition in terms of functions Qλ , which selects the relevant subset
of integral values of λ when one deals with QFT’s on the “true” AdS.
Thermal properties
For all privileged QFT’s on Minkowski, de Sitter and anti-de Sitter spacetimes, a KMS-condition is satisfied in all the families of world-lines contained in
(c)
complex hyperbolae hρ , as described above in subsec 3-1. The corresponding
1
, where ρ
temperature, which is of purely geometric origin, is T = β −1 = 2πρ
is the radius of the hyperbola. It is therefore related to the acceleration of the
corresponding observers through the formulae specified in 3-1 for each case. This
property , which can be called a generalized Unruh effect deserves the following
additional comments.
For the case of de Sitter spacetime, the thermal interpretation covers all
possible timelike orbits of one-parameter subgroups of G, including those of the
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Thermal Aspects in Quantum Field Theory
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inertial motions (namely the meridian hyperbolae of X), for which the associated
1
temperature is minimal and equal to 2πR
.
For the case of anti-de Sitter spacetime (or its covering), the thermal interpretation is limited to accelerated motions whose acceleration a is larger than R1 ,
2
1
1
with a temperature T = 2π
a − R12 2 . For the parabolic trajectories, which correspond to the acceleration a = R1 and to the zero temperature limit, it can be
shown that the corresponding energy spectrum is bounded from below: the spectral condition, postulated for all the elliptic orbits thus extends to the parabolic
ones.
Concerning the proof of this KMS analytic structure in all privileged QFT’s,
one must distinguish the case of free fields and generalized free fields, from the
general case of interacting fields. In the former case, the field theory is completely
determined by its two-point function. So it results from the property of maximal
analyticity of the latter that all the coresponding n−point functions (which are in
that case combinations of products of two-point functions) satisfy maximal analyt(c)
icity and geometrical periodicity in all sections of X (c) by the complex curves hρ .
Therefore all KMS-conditions of the type described in Sec 2 are easily obtained as
a byproduct. In the latter case, the proof of the relevant KMS-conditions for all
the n−point functions on the basis of the general postulates listed in 3-2 is more
technical, since it necessitates the application of an analytic completion procedure,
which we have given in [3,4] under the subtitle “proving the Bisognano-Wichmann
analyticity property”. As a matter of fact, for the case of the Unruh effect in
Minkowski spacetime, the first general proof of the corresponding KMS-property
for general interacting fields has been given in [14] by alternative methods based
on operator algebras. In this connection one should also quote other presentations
and analyses of the generalized Unruh effect in the approach of algebraic QFT,
in particular those of [15] for de Sitter and [16] for anti-de Sitter (see also [17]).
However, we did not pretend to give here a survey of all the investigations and
possible viewpoints concerning the Unruh-type phenomenons. The purpose of our
presentation (and this will be the conclusion to our first part) was to show that
for Minkowski, dS and AdS spacetimes, there holds a unified viewpoint on the
thermal aspects of privileged QFT‘s: it is clearly exhibited by the links between
complex geometry and the global analyticity properties in spacetime variables of
those (free or interacting) theories.
4 Thermal quantum field theories in Minkowski spacetime:
particle aspects
In relativistic quantum field theories with a ground state, (or “theories at zero
temperature”),also called previously “privileged QFT’s in Minkowski spacetime”,
the particle aspect is linked with the occurrence of poles for the Green functions
of the fields in momentum space. In particular, the “elementary particles” of the
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Ann. Henri Poincaré
field Φ(x) (if they exist) correspond to the occurrence of real poles for the twopoint Green function in momentum space, or equivalently of discrete measures
δ(p2 − m2i ) in the corresponding energy-momentum spectrum. Moreover, this is
closely related to the Haag-Ruelle construction of asymptotic free fields Φin (x)
1
and Φout (x) for the interacting field Φ(x). The choice of free propagators (p2 −m
2)
i
for starting perturbation theory is also justified by the latter facts. In thermal
quantum field theories, all these notions must be reconsidered.
After a brief summary of the postulates of thermal QFT, we shall give
i) a general (nonperturbative) study of thermal two-point functions, resulting
in a Källèn-Lehmann-type representation whose weight depends not only on
the mass but also on the space variables x.
ii) a related characterization of the constituent particles of the field in the thermal bath, in terms of thermal free two-point functions accompanied by x−
dependent “damping factors”. The occurrence of such factors is equivalent to
smoothing the usual pole singularity in momentum space
iii) the introduction of “asymptotic generalized free fields” (AGFF) as substitutes
to the usual free fields Φin (x) and Φout (x). These AGFF are associated with
“asymptotic two-point functions” which depend through their damping factor
both on the temperature and on the interacting field.
Explicit nonperturbative computations of the asymptotic two-point functions in
models such as the Φ44 −interaction are feasible. They allow one to give suggestions
for the general type of singularities which might represent thermal particles in
momentum space, as substitutes to the usual poles.
4.1
General results on thermal two-point functions
In the general approach of thermal quantum field theory at temperature T = β −1 ,
one keeps from the relativistic framework the postulate of local commutativity
(or Einstein causality); relativistic algebraic relations such as those which define
the free fields are also preserved. However, the Lorentz symmetry is broken: there
exists a privileged Lorentz frame, determined by the thermal bath corresponding
to a choice of coordinates x = (x0 , x). Time-translation invariance expresses the
stability of thermal states and the spectral condition is replaced by the KMScondition with periodicity iβ in the complex time variable z0 = x0 + iy0 (see Sec
1 and 2). In the framework which we consider, the invariance (of the correlation
n−point functions) under all space-translations and rotations is also maintained.
The general study of the two-point function can be summarized as follows.
Considering for simplicity a single scalar field, one introduces the correlation (or
Wightman) functions W (x) =< Φ(x(1) )Φ(x(2) ) >β (with x = x(1) −x(2) ), W (x) =
W (−x), the commutator function C(x), the retarded and advanced functions R(x)
and A(x), such that:
W − W = C = R − A, with R(x) = θ(x0 )C(x); A(x) = −θ(−x0 )C(x)
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and correspondingly, for the Fourier transforms in momentum space:
W̃ − W̃ = C̃ = R̃ − Ã.
By applying the results described in Sec 1 (KMS-condition), we can say that W and
W are the boundary values on the reals from the respective sides mz0 < 0 and
mz0 > 0 of a function W (z0 , x) holomorphic and iβ− periodic in z0 ; its domain
is the periodic cut-plane generated by the strip −β < mz0 < 0. Moreover the
causality postulate implies the analyticity of W at all points z0 = x0 +iy0 such that
|x0 | < |x|, y0 = inβ (which implies the connectedness of the analyticity domain
of W ).
Equivalently, there holds the following structural properties of the Fourier
transforms in the complexified energy variable k0 = p0 + iq0 (for all real momentum p):
W̃ (p0 , p) =
1
C̃(p0 , p),
1 − e−βp0
W̃ (p0 , p) = −
1
C̃(p0 , p
).
1 − eβp0
)
As usual, the Green functions R̃ and à are boundary values of functions R̃(k0 , p
) of the complex energy k0 = p0 + iq0 , respectively holomorphic in
and Ã(k0 , p
the upper and lower half-planes. However, in contrast to the zero-temperature
case, the relations of the latter with the Euclidean functions are “discretized”: the
) and Ã(k0 , p) at the Matsubara energies k0 = 2niπ
are equal to
values of R̃(k0 , p
β
the spatial Fourier transforms of the Fourier coefficients of the Schwinger function
W (iy0 , x) (periodic in y0 ).
Applying the previous structural properties to the scalar free field with mass
m determines uniquely its thermal two-point function at temperature β −1 as being
given by the following formula
1
1
e−ip·x .
dp ε(p0 )δ(p2 − m2 )
Wm,β (x0 , x) =
3
(2π)
1 − e−βp0
For general two-point functions (of interacting fields) at temperature β −1 ,
the full exploitation of the causality postulate implies the existence of a KällènLehmann-type representation [6], which incorporates the whole analytic structure
previously described and can be written as follows:
∞
W (x0 , x) =
dm Dβ (x, m) Wm,β (x0 , x).
0
In the latter, the weight-function Dβ (x, m) is interpreted as a “damping factor”
which takes into account the dissipative propagation of the field in the thermal
medium. In the usual Källèn-Lehmann representation of the zero-temperature case
(which has a Lorentz-invariant form), this factor is constant with respect to x
and reduces to an m−dependent weight ρ(m), while the thermal free two-point
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Ann. Henri Poincaré
function
free positive-energy Wightman function Wm,∞ (x0 , x) =
is replaced2 by the
1
2 −ip·x
dp
θ(p
)δ(p
−
m
)e
.
3
0
(2π)
Remark It is possible to justify [18] that (as in the zero-temperature case) the
function W has joint analyticity properties in all spacetime variables, as a “remnant of the relativistic spectral condition” (no longer true by itself in the thermal
bath). This “relativistic KMS condition” (which incorporates the previous KMScondition in the variable z0 ) states that W (z) is holomorphic in an (iβe0 −periodic)
analyticity domain, which is generated by the “tube” {z = x + iy; y ∈ V + , βe0 −
y ∈ V + } (where e0 is the timelike unit vector (1, 0)). This analyticity property is
easily checked to be satisfied by the free-field functions Wm,β . Its validity in the
general case is also equivalent to the fact that the corresponding damping factor
Dβ (x, m) is holomorphic in x in the tube domain {z = x + iy; |y | < β2 }.
4.2
Description of constituent particles
By analogy with the zero-temperature case, we shall assume that the presence
of a constituent particle with mass m0 created by the field in the thermal bath
is mathematically characterized by the occurrence of a discrete δ−term in the
damping factor Dβ of the two-point function of that field, namely that one has:
Dβ (x, m) = Zβ (x) δ(m − m0 ) + Dβ,c (x, m),
where Dβ,c is smoother than δ with respect to m. The damping function Zβ (x)
will then represent the effect of the interaction of that constituent particle with
the thermal bath.
The presence of such a discrete term in Dβ corresponds to the manifestation
of an asymptotically dominant contribution at infinite real time x0 in the twopoint function Wβ (x). This contribution, which is given by the distinguished term
(dom)
Wm0 ,β (x) = Zβ (x)× Wm0 ,β (x), enjoys the following asymptotic properties, which
can be considered as physically satisfactory: at fixed x (i.e. at rest), it decreases as
3
|x0 |− 2 (the particle is not submitted to collisions from the thermal bath); along
3
any direction x = v x0 , it has a damped behaviour of the form Zβ (v x0 ) × |x0 |− 2
(the particle is submitted to collisions from the thermal bath).
In energy-momentum space however, the usual “pole-particle” situation is
no longer valid, since the Fourier transform of the previous dominant contribution
(dom)
Wm0 ,β (x0 , x) takes the form of a convolution product, namely:
p − u)2 − m20
ε(p0 )δ p20 − (
1
(dom)
W̃m0 ,β (p0 , p) =
.
du Z̃β (u) ×
(2π)3
1 − e−βp0
The effect of the latter is to wipe out the usual discrete mass shell contribution,
and as a matter of fact, this feature is in agreement with a general theorem of
thermal QFT [19] according to which the existence of a discrete δ in the energymomentum spectrum, corresponding to a particle with a sharp dispersion law, can
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Thermal Aspects in Quantum Field Theory
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only occur in a field theory where the incoming and outgoing fields are the same,
namely a field theory without interaction with the thermal bath.
(dom)
To the previous correlation function Wm0 ,β there corresponds the following
retarded Green function in momentum space:
1
1
(dom)
R̃m0 ,β (k0 , k) =
du Z̃β (u) ×
,
2
(2π)3
k0 − (k − u)2 − m20
which yields after angular integration (assuming rotational invariance):
∞
2
k 2 − u)2 − m2
−
(
1
1
k
(dom)
0
0
R̃m0 ,β (k0 , k) =
u
du
Z̃
(u)
×
log
.
β
(2π)3 0
2 k 2
k 2 − ( k 2 + u)2 − m2
0
0
We shall see below that this formula allows one to maintain the possibility of
analytic continuation of the retarded Green function (from the upper k0 −plane
across the reals) with the occurrence of “momentum space singularities associated
to the constituent particle with mass m0 ”, which however will be generally more
complicated than simple poles.
4.3
Asymptotic generalized free fields: asymptotic dynamics
The fact that no asymptotic free fields can exist in thermal QFT except if the
theory is itself without interaction [19], together with our previous analysis of
the asymptotically dominant behaviour of thermal two-point functions in the time
variable have led us to introduce the concept of asymptotic generalized free field or
asymptotic dynamics [7], by which we mean that at asymptotic times, the presence
of the thermal bath maintains a manifestation of the dynamics of the interacting
field, which is of course temperature-dependent and vanishes only in the limit of
zero-temperature.
Time-clustering assumption We are led to make an assumption on the asymptotic
behaviour of the thermal n−point functions of the field < Φ(x(1) ) . . . Φ(x(n) ) >β ,
when the minimal internal time-interval ∆ = inf{|x(j)0 −x(k)0 | : j, k = 1, . . . , n, j =
k} of the configuration (x(1) , . . . , x(n) ) tends to infinity. We assume that, as in
the usual construction of asymptotic fields (although not in the same limiting
procedure), this asymptotic behaviour is dominated by the terms containing the
maximal number of two-point functions in the expansion of < Φ(x1 ) . . . Φ(xn ) >β
in terms of its truncated (or connected) parts. In heuristic terms, this assumption means that there hold no collective memory effects in the underlying KMS
state. (Note that in a satisfactory formulation, the exclusion of possible low-energy
excitations must be performed at first by making use of some appropriate regularization of the field in the time-variable [7]). Moreover, the two-point functions
are themselves supposed to be asymptotically dominated in time by a constituent(dom)
particle contribution Wm0 ,β (x0 , x) of the form specified above, involving an unknown damping factor Zβ (x).
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Ann. Henri Poincaré
So the timelike asymptotic description of an interacting field theory in a
thermal state | >β appears to be given by a thermal generalized free field Φ0 (x),
(dom)
entirely characterized by its two-point function Wm0 ,β (x) = Zβ (x) × Wm0 ,β (x).
It turns out that the determination of Zβ (x) in a given specific field model such
as the (hypothetic) Φ44 −model is feasible in a simple way, on the following basis.
3
At fixed x, Φ0 (x0 , x) behaves as |x0 |− 2 . Now the equation of the interacting field
.
(of the form N (x) = ∂µ ∂ µ Φ(x) + m20 Φ(x) + g”Φ3 (x)” = 0) should only be satisfied
in an asymptotic form by the asymptotic field Φ0 (x), which can be reasonably
understood as follows.
.
The field-function N0 (x) = ∂µ ∂ µ Φ0 (x) + m20 Φ0 (x) + gΦ30 (x) should be such
3
that at fixed x, the product |x0 |− 2 × N0 (x0 , x) tends to zero for |x0 | tending to
infinity. (In writing that condition, one can give a meaning to the third power of
this noninteracting field in a standard way). It turns out that the latter condition
allows one to determine Zβ (x) in a fully consistent way as a solution of the following
Laplace-type equation:
(−∆ + 3gk(β))Zβ (x) = 0,
where the (finite) quantity
.
k(β) = 2(2π)−3
dpθ(p0 )δ(p2 − m20 )(eβp0 − 1)−1
tends to zero for T = β −1 tending to zero. One then obtains the following results:
a) for g < 0:
sin(κ(β)|x|)
δ(u − κ(β))
, Z̃β (u) =
,
Zβ (x) =
κ(β)|x|
κ(β)2
b) for g > 0:
Zβ (x) = Cst
e−κ(β)|x|)
,
|x|
Z̃β (u) =
Cst
,
u2 + κ(β)2
1
where κ(β) = [3|g|k(β)] 2 .
4.4
Conclusions
The type of result which we obtain for Z̃β (u) in the previous model indicates that
(dom)
(real or complex) singularities of the Green function R̃m0 ,β (k0 , k) can be generated
by those of Z̃β (u) through the integral representation written above (at the end
(dom)
of 4-2). More precisely, one sees that these singularities of R̃m0 ,β should always
be localized on surfaces of toroidal shape with equation
k02 − ( k 2 ± a)2 − m20 = 0,
where a can be either real or complex (in the example of 4-3, a was equal to κ(β)
for g < 0 and to iκ(β) for g > 0). Moreover these singularities should be in general
Vol. 4, 2003
Thermal Aspects in Quantum Field Theory
S879
of logarithmic rather than polar type, and should also be accompanied by real cuts
starting at k0 = ±m0 .
It is worth noticing that our proposal for characterizing the momentum space
singularities of the particles in thermal quantum field theory is substantially different from other recent proposals [20] where complex pole-approximations of the
1
form (k0 −E(p))(k
p) analytic) have been chosen as substitutes to
∗ p)) (with E(
0 +E (
the usual real poles as a starting point for perturbation theory. While for the standard case of zero-temperature quantum field theory the formalism of perturbation
theory is in agreement with all concepts and results of the general field-theoretical
framework (discrete energy spectrum, asymptotic free fields, general form of the
two-point functions, pole-particle concept), the situation is quite different for the
case of thermal field theory. As shown above, all the previous concepts and results are substantially modified (no discrete spectrum, no asymptotic free-fields
but “generalized free-fields and asymptotic dynamics”, complex singularities of
toroidal shape and smoother than poles to be associated with particles): this suggests that a “reasonable” starting point for an analytic perturbation formalism in
momentum space should already take into account the form of the interaction in
the approximate form of the propagator.
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Nucl. Phys. B 627, 289 (2002).
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Ann. Henri Poincaré
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58, 105002 (1998); “Analytic properties of finite-temperature self-energies”,
hep-ph/0203057.
Jacques Bros
PhT
CEA-Saclay
F-91191 Gif-sur-Yvette cedex
France