数理解析研究所講究録
1142 巻 2000 年 123-136

123

Towards Local Temperature States in QFT*
Izumi OJIMA
Research Institute for Mathematical Sciences,
Kyoto University, Kyoto 606-8502, Japan

Simple example of local temperature state

1

Aiming at a general formulation of non-equilibrium states in the frameworkof QFT, we start from a simple model-example of local temperature state:
be a massless free scalar field (in four dimensions) characterized byLet$\varphi(x)$$\square \varphi$$[\varphi(x), \varphi(y)]$

$=$

$0$

$=iD(x-y)= \int\frac{d^{4}p}{(2\pi)^{3}}e^{-ip(x-y)}\epsilon(p_{0})\delta(p^{2})$Given a two-point function$\omega$of$\varphi$(1),$\omega^{(2)}(\varphi(x)\varphi(y))$.

(2), we can define a quasi-free statethrough the Wick formula:$\omega(\varphi(x_{1})\varphi(x_{2})\cdots\varphi(x_{r}))$:,$\sum_{pairings}\omega^{(2)}(\varphi(x_{i_{1}})\varphi(x_{i_{2}}))\cdots\omega^{(2)}(\varphi(x_{i_{f-1}})\varphi(x_{i_{\gamma}}))$$(\mathrm{i}\mathrm{f} r:\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n})_{(3)}$$=\{$(if : odd).

$0$

$r$

When a two-point function$\omega^{(2)}$is chosen consistently with Eqs. $(1, 2)$ as$\omega_{\beta}^{(2)}(\varphi(x)\varphi(y))=\int\frac{d^{4}p}{(2\pi)^{3}}e^{-ip(x-y)}\epsilon(p_{0})\delta(p^{2})\frac{1}{1-e^{-\beta p_{0}}}$,(4)

describes a global thermal equithe corresponding quasi-free stateanypair of polynomial fields $A=$for
condition
librium satisfying the KMS,
and$\omega=\omega_{\beta}$$\varphi(x_{1})\cdots\varphi(x_{r})$$B=\varphi(y_{1})\cdots\varphi(y_{s})$$\omega_{\beta}(A\alpha_{(i\beta,\mathrm{O})}(B))=\omega_{\beta}(BA)$*Talk based on a joint work of D.Buchholz, H.Roos and,$\mathrm{I}.\mathrm{O}$(5). in progress

124

where
, such a choice of two-point function asof$\alpha_{(i\beta,\mathrm{O})}(B):=\varphi(y_{1}+(i\beta, 0))\cdots\varphi(y_{s}+(i\beta, 0))$(symbolically). In place$\omega_{\beta}^{(2)}$(6)$\omega^{(2)}(\varphi(x)\varphi(y))=\int\frac{d^{4}p}{(2\pi)^{3}}e^{-ip(x-y)}\epsilon(p_{0})\delta(p^{2})\frac{1}{1-e^{-\beta^{\mu}(\frac{x+y}{2})p_{\mu}}})$is also allowed consistently with Eqs. $(1, 2)$ if a spacetime-dependent inversetemperature four-vector is given by. In this case we canverify$\beta^{\mu}(x)=\beta^{\mu}+\gamma_{\nu}^{\mu}x^{\nu 1}$$(7)$$\omega^{(2)}(\varphi(x)\varphi(y+i\beta(\frac{x+y}{2}))=\omega^{(2)}(\varphi(y)\varphi(x)),$which can be interpreted as a localized version of KMS condition. Contrary to
the case of the global equilibrium , however, it is not possible to generalize
this relation to -point functions with $n\geq 3$ in the similar form to:$\omega_{\beta}$$n$$\mathrm{E}\mathrm{q}.(5)$e.g.,$\omega(\varphi(x_{1})\varphi(x_{2})\cdots\varphi(x_{r})\varphi(y_{1}+i\beta(\frac{x_{1}+y_{1}}{2}))\varphi(y_{2}+i\beta(\frac{x_{2}+y_{2}}{2}))\cdots$...

$\varphi(y_{r}+i\beta(\frac{x_{r}+y_{r}}{2})))$$\neq\omega(\varphi(y_{1})\varphi(y_{2})\cdots\varphi(y_{r})\varphi(x_{1})\varphi(x_{2})\cdots\varphi(x_{r}))$,(8)

just because of the spacetime dependence of. Namely, the similarity ofthe state given by Eqs. $(6, 3)$ to a global thermal equilibrium
holds only
up to two-point function. In this way, can be taken as a model example
of a local temperature state such that it is only locally in equilibrium in thesense of.
Remark: The breakdown of KMS condition for -point functions with $n\geq$
is natural as a signal of non-equilibrium, in view of the zeroth law ofthermodynamics which claims the validity of transitive law in the thermal
equilibrium contact relations of two bodies: To inspect for the breakdown oftransitivity due to non-equilibrium, we need to examine the relation among
three bodies.
Generalized Stefan-Boltzmann law and “Local Thermometers”
Now, a remarkable property of this state is found in the following formula:$\beta^{\mu}(x)$$\omega$$\omega_{\beta}$$\omega^{(2)}$$\omega$$\mathrm{E}\mathrm{q}.(7)$

$n$

$3$

$\omega$

1Because of the positivity condition to be satisfied by
and

$y$

may not cover all the spacetime$\mathbb{R}^{4}$.

$\omega^{(2)}$, the region allowed for

$x$

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$\lim_{\xiarrow 0}$$=$$\omega(\partial_{\mu_{1}}^{\xi}\partial_{\mu_{2}}^{\xi}\cdots\partial_{\mu,}^{\xi} :\varphi(X-\frac{\xi}{2})\varphi(X+\frac{\xi}{2}):)$$\lim_{\xiarrow 0}[\omega(\partial_{\mu_{1}}^{\xi}\partial_{\mu_{2}}^{\xi}\cdots\partial_{\mu_{r}}^{\xi}\varphi(X-\frac{\xi}{2})\varphi(X+\frac{\xi}{2}))$$- \omega_{\mathrm{v}\mathrm{a}\mathrm{c}}(\partial_{\mu_{1}}^{\xi}\partial_{\mu_{2}}^{\xi}\cdots\partial_{\mu_{\Gamma}}^{\xi}\varphi(X-\frac{\xi}{2})\varphi(X+\frac{\xi}{2}))]$$=$$T(X)^{r+2}C_{\mu_{1}\mu_{2}\cdots\mu,}$

,

(9)

where
is a vacuum state corresponding to $\beta=\infty$ andconstant tensor vanishing for. $T(X)$ defined by$\omega_{\mathrm{v}\mathrm{a}\mathrm{c}}$$C_{\mu_{1}\mu_{2}\cdots\mu_{f}}$is a$r=\mathrm{o}\mathrm{d}\mathrm{d}$(10)$T(X):=1/\sqrt\overline{\beta^{\mu}(X)\beta_{\mu}(X)}$is a local temperature. In the case of $r=2$ the left-hand side of this formula can easily be related to the energy density with a suitable tensorial
combination, in view of which it gives a generalization of the well-known, for the radiation energy . In the sense of
Stefan-Bolzmann law,
, the operator:: works as a “local$T(X)$
thermometer” to measure a local temperature
at the centre-of-mass
pointof very close two points
andin the limit. Further the totality of operators of this sort are seen to give rise on
right-handtheside to all the even powers of local temperature $T(X)$ (times
some Lorentz tensors of even degree) which generate an algebra of functionsof non-negative temperatnres $T(X)\geq 0$ .
$e=\sigma T^{4}$$\mathrm{E}\mathrm{q}.(9)$$\partial_{\mu_{1}}^{\xi}\partial_{\mu_{2}}^{\xi}\cdots\partial_{\mu_{r}}^{\xi}$$e$$\varphi(X-\frac{\xi}{2})\varphi(X+\frac{\xi}{2})$$x=X- \frac{\xi}{2}$$X= \frac{x+y}{2}$$y=X+ \frac{\xi}{2}$$\xiarrow 0$

2

Definitions of generalized and local thermal
statesMotivated by this example, we explore the possible framework for accommodating some class of states describing local thermal situations: Basic idea
is to compare a given state in a small neighbourhood of a spacetime pointwith all the KMS states at possible temperatures by means of certain setof point-like local observables playing the role of “thermometers”. For thispurpose, we need the following definitions.
Definitions:(i) Set of thermal states $K$ is defined by the closed convex hull of all the, whereKMS states,is the set of KMS states$K:=\overline{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}(\bigcup_{\beta\in \mathbb{R}_{+}}K_{\beta})}$at inverse temperature .
$\beta$$K_{\beta}$

126

with
: The relativistic version can be obtained by replacing. Depending on theforward lightcone, where
situations, we freely move from one version to another.
$NB$$\beta\in \mathbb{R}_{+}$$\beta^{\mu}\in V_{+}:=$$\beta=\sqrt{\beta^{\mu}\beta_{\mu}}$

is called aof (point-like) local observables(ii) A set
thermometer set or a iscriminating set if
implies. The linear hull of is denoted byfor. Although the identity operator 1 is irrelevant to the
purpose of discriminating different states, we understand by convention
that it belongs to.
$A_{n}$$\mathcal{T}=\{A_{n};n\in \mathrm{N}\}$$\omega(A_{n})=\omega’(A_{n})(\forall n\in \mathbb{N})$$d.$$\omega=\omega’$$\omega,\omega’\in K$$\mathcal{T}$$\mathcal{L}:\mathcal{L}:=\mathrm{L}\mathrm{i}\mathrm{n}\mathcal{T}$$\mathcal{T}:1\in \mathcal{T}$(iii) $K$ -normof a local observable is defined by $||A||_{K}:= \sup_{\omega\in K}|\omega(A)|$ .
For the sake of simplicity, we assume the absence of phase transition,.
, in which case
for
i.e.,
$A$

$||A||_{K}$$K_{\beta}=\{\omega_{\beta}\}$$||A||_{K}= \sup_{\beta\in V_{+}}|\omega_{\beta}(A)|$$\forall\beta\in V_{+}$

is called a K(iv) A local observable satisfying $\omega(A)\geq 0$ for$K$
cone de-positiaelement, the totality of which constitutenoted by $\prime P_{K}:=\{A;\omega(A)\geq 0\forall\omega\in K\}=\{A;\omega_{\beta}(A)\geq 0\forall\beta\in V_{+}\}$.
, the set of $K$ -positive thermometers.
We also denote$A$$\forall\omega\in K$$positi\mathrm{t}^{\Gamma}e$$\mathrm{r}^{\gamma}e$$Pc:=\prime P_{K}\cap \mathcal{L}$(v) The set of thermal fun ctions is defined bywith$\mathcal{F}_{0}:=\mathrm{L}\mathrm{i}\mathrm{n}\{f_{n}$$\omega_{\beta}(A_{n}),$$A_{n}\in \mathcal{T},$$n\in \mathbb{N}\},$$\mathcal{F}:=\overline{\tau_{0}}^{||\cdot||_{\infty}}$$||\cdot||_{\infty}$;$f_{n}(\beta)$$:=$

being the supremum

norm.
Further, we impose the following two restrictions on the thermometer set:$\mathcal{T}$is linearly independent,.
is dense in
Rom these it immediately follows that
because offor any spacetime translations
implies,the relation$-\Phi$ :defines an isometric 1-1 map of onto , whereis equipped with the supremum norm, and with the $K$ -norm-$\{f_{n}\}$-$F_{0}$-$A_{n}\in \mathcal{T}$$C_{0}(V_{+})$$\alpha_{x}(A_{n})\not\in \mathcal{T}$$\alpha_{x}$$\omega_{\beta}\circ\alpha_{x}=\omega_{\beta}$$A_{n}\in \mathcal{T}\mapsto f_{n}\in F_{0}$$\mathcal{L}$$\mathcal{F}_{0}$$||\cdot||_{K}.2$$\mathcal{L}$$\mathcal{F}_{0}$Criteria for generalized thermal states:
A state is called a generalized thermal state w.r.t. a thermometer set(“ -GTstate” or “GTstate”, in short) if it is $K$ -bounded and K-positiveon in the following sense:$\mathcal{T}$$\omega$$\mathcal{T}$$\mathcal{L}$$| \omega(\sum c_{n}A_{n})|\leq C||\sum c_{n}A_{n}||_{K}$;$\omega \mathrm{r}_{P_{\mathcal{L}}}\geq 0$

2Of course, the -norm does not define a Hausdorff topology on .
$K$$\mathcal{L}$

.

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The physical meaning of this criterion is seen in the following result:Lemma: A generalized thermal state defines a positive linear functional, and extenon , i.e.,byis represented by a probability measuresion by continuity. Therefore,onin such a form as$\omega$$\varphi_{\omega}:=\omega 0\Phi^{-1}$$\varphi_{\omega}\in \mathcal{F}^{*}$$\varphi_{\omega}(\sum c_{n}f_{n})=\omega(\sum c_{n}A_{n})$$\mathcal{F}_{0}$$\mu_{\omega}$$\varphi(v$$\mathbb{R}_{+}$$\omega(\sum c_{n}A_{n})=\varphi_{\omega}(\sum c_{n}f_{n})=\int d\mu_{\omega}(\beta)\sum c_{n}f_{n}(\beta)=\int d\mu_{\omega}(\beta)\omega_{\beta}(\sum c_{n}A_{n})$.
in short:
Namely, a generalized thermal state is a state which can be approximatedaround a spacetime point by a statistical average over thermal equilibrium
state is called a local temperature state (or LT
states. More strongly, a.
is concentrated on a single point
state, in short) if$\omega \mathrm{r}_{c}=\int d\mu_{\omega}\omega_{\beta}\mathrm{r}_{c}$$\mathrm{c}^{r}\mathrm{r}$$\omega$$\beta\in \mathbb{R}_{+}$$\mu_{\omega}$

Remarks:i) It is due to our convention ofa probability measure:$1\in \mathcal{T}$that the measure$\mu_{\omega}$

is normalized as$1= \omega(1)=\int d\mu_{\omega}(\beta)\omega_{\beta}(1)=\int d\mu_{\omega}(\beta)$.

ii) Any state $\omega\in K$ (given as convex combination of KMS states) is a
generalized thermal state w.r.t. any , since $\omega(B)>0\forall B\in P_{K}$ .
$Pc$ .
Therefore, the notion of GT states defined above actually givea generalization of thermal equilibrium states in a three-fold way, intheir being i) relativistic, ii) mixtures of different temperatures, andiii) localized in small neighbourhoods of spacetime points.
$\mathcal{T}$$\supset$iii) “Point-like local observables” belonging to can be formulated in thecontext of local nets as the objects dual to state germs (in the sense of[1] and [2] .
$\mathcal{T}$$)$iii) When a $K$ -bounded state is allowed to be non-K-positive, thenbecomes a signed measure.
$\omega$iv)$\omega^{(\mu)}:=\int d\mu\omega_{\beta}$

may be a$\mathcal{T}- \mathrm{G}\mathrm{T}$state even if$\mu$$\varphi_{\omega}$

is not positive.

Desired localization properties of : For any spacetime point , there
consisting of local observables belonging (more apis a thermometer set
where is an arbitrary small neighbourhood
propriately, affiliated) toof (“ concentrated at ”).
Existence of local thermometer sets: Assume (i) weak additivity, (ii)for boundedregions , then there is anorm continuity of) for an arbitraryA(orthermometer set$x$$\mathcal{T}$$\mathcal{T}$$A(\mathcal{O})$$\mathcal{O}$$x$$\mathcal{T}$$\mathcal{O}$$x$$\mathcal{O}$$\betarightarrow\omega_{\beta}\mathrm{r}_{A(O)}$$\mathcal{T}\subset A(\mathcal{O}_{0})$$\mathcal{T}\subset\{A;$$\eta A(\mathcal{O}_{0})\}$

;

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small

$\mathcal{O}_{0}$.

Lemma: Letbe concentrated at; furthermore, letbe a$N$T-GT state for in a neighbourhood of , i.e., is locally a generalized
thermal state at the point , defining a corresponding measure . Thenis weakly continuous, i.e.,
depends continuously on $x\in N$ forall.
Especially, if the mean temperat at is defined
,
$N$
then
is continuous in .
$0\in \mathbb{R}^{4}$$\mathcal{T}$$\omega\circ\alpha_{-x}$

$0$

$x$

$\omega$$x$$\mu_{x}$$\mu_{x}$$\int d\mu_{x}(\beta)F(\beta)$$F\in C_{0}(\mathbb{R}_{+})$$\mathrm{u}re$$\mathrm{b}\mathrm{y}\overline{T}(x):=\int d\mu_{x}(\beta)\beta^{-1}$$x$$xrightarrow\overline{T}(x)$2.1

Choice of thermometers: Equivalence classes of$A\in \mathcal{T}$We define an equivalence relation $A\sim B$ between $A,$
by $\omega(A)=$forand denote the equivalence class of by $[A]$ . It is clearthat the $K$ -norm and the $K$ -positivity are independent of the choice of therepresentatives in. However, the validity of the above criteria for-states will in general depend on the choice of specific representatives from
. Although this point need be further elaborated, the best choice seems
to be given by considering “derivatives conserving the center of mass”:.
. To justify it as a reasonable choice, we recall herethat the requirement of linear independence offorbids the operation oftranslations
on elements of : i.e.,
implies, because. In view of the relation for two-point product operators$B\in \mathcal{T}$$\omega(B)$$\forall\omega\in K$$A$$[A_{n}],$$n\in \mathrm{N}$$\mathrm{G}\mathrm{T}$$[A_{n}]$$\mathrm{e}.\mathrm{g}$$\partial_{\mu}^{\xi}(\Phi(\xi/2)\Phi(-\xi/2))\int_{\xi=0}$$\mathcal{T}$$\alpha_{x}$$\mathcal{T}$$A_{n}\in \mathcal{T}$$\alpha_{x}(A_{n})\not\in \mathcal{T}$$\varphi(x)\varphi(y)$$\omega_{\beta^{\circ}}\alpha_{x}=\omega_{\beta}$$\frac{d}{dt}\alpha_{le_{\mu}}(\varphi(X-\frac{\xi}{2})\varphi(X+\frac{\xi}{2}))|_{t=0}=\partial_{\mu}^{X}(\varphi(X-\frac{\xi}{2})\varphi(X+\frac{\xi}{2}))$,

this amounts in its infinitesimal version to excluding derivatives
w.r.t.
the centre-of-mass coordinates. In fact, the derivative w.r.t. therelative coordinates $\xi=y-x$ is sufficient to generate the whole thermometer
set in our previous discussion of local temperature states in the massless freescalar field. In general, we will be led to the choice of operators in such aform as$\partial_{\mu}^{X}$$X= \frac{x+y}{2}$$\partial_{\mu}^{\xi}(\Phi_{1}(X+\gamma_{1}\xi)\Phi_{2}(X+\gamma_{2}\xi)\ldots\Phi_{n}(X+\gamma_{n}\xi))\mathrm{r}_{\epsilon=0}$$\partial_{\xi}$with $\sum\gamma_{i}=0$ .

Concerning this choice we can see its deeper meanings in more general contexts such as the classificaCion of states from the viewpoint of singularity
and also as the extension of the locaUy thermal notions to curved spacetime
situations:

129i) The data for classifying different spieces of states, such as vacuum, temperature states, and so on, can be provided by quantities defined on
the cotangent spaces of spacetime which are closely related with theand relative onesinseparation into centre-of-mass coordinates
the case of two-point functions. For instance, in the discussion ofmassless free scalar field model of local temperature states, we have(:seen that the “normal product” relative to a vacuum,,
plays an important role. This takes care of the separation of the most
singular and less singular terms atas: Namely, rewriting$X^{\mu}$$\xi^{\mu}$$\rho(X, \xi)\equiv\omega$$\varphi(X-\frac{\xi}{2})\varphi(X+\frac{\xi}{2}):)=\omega(\varphi(X-\frac{\xi}{2})\varphi(X+\frac{\xi}{2}))-\omega_{\mathrm{v}\mathrm{a}\mathrm{c}}(\varphi(X-\frac{\xi}{2})\varphi(X+\frac{\xi}{2}))$$\xiarrow 0$$\mathrm{E}\mathrm{q}.(9)$$\omega(\varphi(X-\frac{\xi}{2})\varphi(X+\frac{\xi}{2}))=\omega_{\mathrm{v}\mathrm{a}\mathrm{c}}(\varphi(X-\frac{\xi}{2})\varphi(X+\frac{\xi}{2}))+\rho(X, \xi)$

,

(11)

we note that the vacuum term on the RHS gives the most singularcontribution in the limit. In this sense, thermal aspects can betaken as those concerning the next-to-leading contributions at a point$\xiarrow 0$$X\mathrm{w}_{4}\mathrm{i}\mathrm{t}\mathrm{h}\xi\neq=0$.

ii) This can be seen more clearly in the microlocal spectrum condition characterizing the vacuum states in curved spacettime (see e.g., [3]). Here,the condition for vacuum states is formulated in terms of wave front
set which specifies the locations of singularities of distibutions in the
cotangent bundle (more precisely, sphere bundle) of spacetime manifold. While we do not need the details of the microlocal spectrum
condition here, its essence lies in the condition imposed upon the wave
front sets of -point functions from the viewpoint of paraUel transportability of momenta between vertices and it is interesting to notethat this condition gives a close relationship between the characterization of states and the singularities of Wightman functions in cotangent
spaces.
$n$, of a distribution $\phi\in D’(V)$Remark: The wave front set,
) is defined in(Vas the complement of the set$(\xi’,p’)$satisfying the property that there exist some neighof points
bourhood of and some conic neighbourhood of such that, for$\mathrm{W}\mathrm{F}(\phi)$$\subseteq \mathbb{R}^{d}$$V\cross(\mathbb{R}^{d}\backslash \{0\})$$U$$\Sigma$$\xi’$$p’$$\forall f\in C_{0}^{\infty}(U),\forall N\in \mathrm{N}\cup\{0\},\forall p\in \mathbb{R}^{d}\backslash \{0\}$$p\in\Sigma\Rightarrow|<\phi,$ $e^{-i<\cdot,p>}f>|\leq C_{f,N}(1+|p|)^{-N}$

holds with some constant. Here, : conic, means that $p\in\Sigma$
implies $tp\in\Sigma$ for all $t>0$ . If does not contain any singularity,
the validity of the above inequality can easily be understood by the$\Sigma$$C_{f,N}$$\phi$

130

Paley-Wiener theorem. This growth condition in the limit of $parrow\infty$
in momentum space just corresponds in coordinate space to the limit
because of the arbitrariness of smooth functionsofin the neighbourhood of .
$f\in C_{0}^{\infty}(U)$$\xiarrow\xi’$$\xi’$

withiii) If we apply this notion to our distribution$\xiarrow0$
corresponds to
as its argument, the most singular term atEq.(ll). Once
of
on
RHS
the vacuum term
remaining as the less singular term is smooth
we remove it,(or more suitably, analytic according to the result of [4]) in . In
the massless free scalar field model of local temperature states, wetogether with its -derivatives
have seen that the function
are sufficient for identifying a local temperature $T(X)$ . The Fourier, (in $parrow\infty$ )transform of this function,
tion defined on the otangent space at $\xi=0$ in
can be viewed as arelative coordinates.
$\omega(\varphi(X-\frac{\xi}{2})\varphi(X+\frac{\xi}{2}))$$\xi$$\omega_{\mathrm{v}\mathrm{a}\mathrm{c}}(\varphi(X-\frac{\xi}{2})\varphi(X+\frac{\xi}{2}))$$\rho(X, \xi)$$\xi$$\xi\vdash+\rho(X, \xi)$$\xi$$p rightarrow\int d\xi e^{ip\xi}\rho(X,\xi)=\tilde{\rho}(X,p)$fun.c$\mathrm{c}$w.r.t. relative coordinates (acting in theiv) Applying the derivative
fiber-direction of the cotangent bundle) corresponds to multiplicationin momentum space. If the relevant functions in the above iii)
byand its
are always ensured to be analytic, then (X, )derivatives will exhaust all the necessary information as (the expectation value of) the discriminating set .
$\xi$$\partial_{\mu}^{\xi}$$p_{\mu}$$p$$-\rangle\tilde{\rho}(X,p)$$\xi-$$\mathcal{T}$v) The above formulation based upon quantities in cotangent bundles canbecome crucial in the attempt of extending basic ingredients in locallythermal situations to curved spacetimes. While the reference system$K$ of thermodynamic equilibria is treated in our formulation as global
KMS states, the roles of these equilibrium states lie in assigning a temperature or temperature distribution to each spacetime point in referprepared in its small neighbourhood.
ence to the thermometer set
Thus, with the aid of normal coordinates and the exponential map(which maps flat tangent spaces onto curved spacetime), it should bepossible to reformulate $K$ as an object living in the tangent bundle ofspacetime, if we restrict our spacetime manifold to he one admitting
complexification (which is crucial for the formulation of KMS condition or its relativistic generalization due to [4] . In the context of thiswill
generalization, the 4-vector nature of inverse temperature
be very interesting in relation with its role to specify the rest frame ateach spacetime point and also its close relationship with the 4-vector. Before discussing the thernature of entropy density current
modynamic behaviours of generalized thermal states in terms of these$\mathcal{T}$$\dot{\mathrm{t}}$$)$$\beta_{\mu}(X)$$s_{\mu}(X)$131

local densities, we need first to examine more closely the thermostatics
in the set $K$ of all the KMS states in order to extend the mutual relations of these thermal quantities from equilibrium regions to those of
our generalized thermal states.

3

Thermostatics in

3.1

Definition of local rest frame and equivalence principle$\mathrm{K}$the relativistic viewpoint, inverse temperature and entropy densityand , respectively: In a refneed be understood as Lorentz four-vectors,$S$
erence frame , let denote the relative velocity of rest frame of our thermal$\mathrm{I}*\mathrm{o}\mathrm{m}$$s$$\beta$$s^{\mu}$$\beta^{\mu}$$\vec{v}$$4- \mathrm{v}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{d}\mathrm{b}\mathrm{y}u^{\mu}=(\frac{\mathrm{m}_{1}(\mathrm{a}\mathrm{t}}{\sqrt{1-|^{arrow}v|^{2}}},\frac{\mathrm{m}\mathrm{p}\mathrm{e}\vec{v}}{\sqrt{1-|\vec{v}|^{2}}}),u^{\mu}u_{\mu}=1).\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{n}\mathrm{i}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{e}\mathrm{t}\mathrm{e}\mathrm{m}- \mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{a}1\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{b}\mathrm{r}\mathrm{i}\mathrm{u}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}T),\mathrm{a}\mathrm{n}\mathrm{d}u^{\mu}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g}$with $\tau:=k_{B}T$
perature has to be treated as a four-vector(formally this is understood as the generalization of the Boltzmann factor$\exp(-\beta E)$ to the invariant expression. The same holds trueof the entropy density: it is to be considered as the four-vector
is the density of equilibrium entropy. (This allows to generalize
where
)
to
the expression
point inIf one can find the local rest frame of the system at eachthe present context of generalized thermal states, the essence of equilibrium$\beta^{\mu}=\tau^{-1}u^{\mu}[5]$$\exp(-\beta^{\mu}p_{\mu}))$$s^{\mu}=s_{\mathrm{e}\mathrm{q}}u^{\mu}$$s_{\mathrm{e}\mathrm{q}}$$\tau^{-1}s_{\mathrm{e}\mathrm{q}}$$\beta^{\mu}s_{\mu}.$$\mathrm{s}\dot{\mathrm{p}}\mathrm{a}\mathrm{c}\mathrm{e}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}$thermodynamics should be reproduced there locally and the descriptions ingeneral frames can be derived kinematically through the Lorentz transforwhich relate the former tomations parametrized by the relative velocity
the latter. This is just Einstein’s equivalence principle applied to thermody$u^{\mu}$namics.

3.2

Thermodynamic relations in equilibrium

To develop machinery for analyzing the general thermal states, we aim here at
extracting the useful essence from thermodynamic relations valid for equilibrium states in the rest frame in order to extend it to their arbitrary mixtures(belonging to the set $K$ ) and to arbitrary Lorentz frames.
is characterized by
In general Lorentz frame, an equilibrium state$\beta\inV_{+}[4]$, and hence, ourafixed
the relativistic KMS condition w.r.t.
set $K$ of all the KMS states viewd from such a frame can be expressed by$\omega_{\beta}$

132. We try to extend the notions of entropy density,
energy density, free energy density, and possibly temperature to all states
belonging to $K$ starting from a rest frame characterized by the relations$u^{\mu}=(1,0),$$s^{\mu}=su^{\mu}=(s, 0)$ .
For a system in equilibrium in the rest frame, let
be the equilibrium entropy density; it is a concave function of the energy
density and of other conserved quantities, indicated by the dots, such as
baryon number density , lepton number density , electric charge density, etc. In the sequel, we shall mostly disregard all other variables than ; ourconsiderations can easily be extendend to the general case. The quantities
and refer to the rest frame of the system under consideration. The
corresponding temperature $\tau(:=k_{B}T)$ is given as usual by.
For an equilibrium state
we have$K:=\overline{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}\{\omega_{\beta}\cdot,\beta\in V_{+}\}}$$\beta^{\mu}=\beta u^{\mu}=(\beta, 0),$$s(\omega_{\beta})\equiv s_{\mathrm{e}\mathrm{q}}(e, \cdots)$$e$$n_{B}$$n_{L}$$e$$q$$s_{\alpha_{1}}$$e$$\tau^{-1}=\partial s_{\infty_{1}}/\partial e$$\omega_{\beta}$$u_{\mathrm{t}\mathrm{h}}^{\mu}(\beta)s_{\mu}(\omega_{\beta})=s_{\mathrm{e}\mathrm{q}}(e_{\beta})$,$\sqrt{\beta^{2}}=\tau^{-1}=\frac{\partial s_{\alpha_{1}}}{\partial u}(e_{\beta})$

,

(12)

with

,

$e_{\beta}=u_{\mathrm{t}\mathrm{h}}^{\mu}(\beta)u_{\mathrm{t}\mathrm{h}}^{\nu}(\beta)T_{\mu\nu}(\omega_{\beta})$

(13)

where
is the energy momentum tensor evaluated in theequilibrium state . (Contrary to some treatments of relativistic thermodynamics, the rest mass is included in theenergy density defined
above). Relations (12) and (13) follow from the invariance of the expressions
by evaluation in the rest system.
$T_{\mu\nu}(\omega_{\beta}):=\omega_{\beta}(T_{\mu\nu})$$\omega_{\beta}$$(” \mathrm{i}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{r}" )$$(u_{\mathrm{t}\mathrm{h}}=(1,0,0,0))$A general stateis of the form
with a probability
measure on . Denoting the average of a function
on $K$ w.r.t. thisprobability measure by, we see that the energy density$\omega\in K$$\rho$$g(\beta)$$V_{+}$$\rho$

in$\omega\in K$$\omega=\int_{V}+d\rho(\beta)\omega_{\beta}$$\langle g\rangle_{\omega}:=\int d\rho(\beta)g(\beta)$

is given by$e(\omega)=\langle e_{\beta}\rangle_{\omega}\equiv\langle e(\omega_{\beta})\rangle_{\omega}$

We define the entropy of$\omega$.

by$s^{\mu}( \omega)=\int_{V^{+}}d\rho(\beta)s^{\mu}(\beta)$

.

(14)

This is a natural definition in view of the fact that two KMS statesandin infinite systems are disjoint for[Bratteli-Robinson II], which
implies that the entropy is additive:.
$\omega_{\beta}$$\omega_{\beta’}$$\beta\neq\beta’$$s(\lambda_{1}\omega_{\beta}+\lambda_{2}\omega_{\beta’})=\lambda_{1}s(\omega_{\beta})+\lambda_{2}s(\omega_{\beta’})$133

Concerning the question as to how the mean temperature of the stateshould be defined, there are a few possibilities such as$\omega=\int_{V^{+}}d\rho(\beta)\omega_{\beta}$$\beta(\omega):=\langle\beta\rangle_{\omega}$,or$\beta(\omega)^{-1}=T(\omega):=\langle\beta^{-1}\rangle_{\omega}$.

A better, and more natural definition is given by$\beta(\omega):=\frac{\partial s}{\partial e}(e(\omega), \ldots)$

.

(15)

:
Rom covariance consideration we can determine the form of
may depend on other conserved quantities as well,
The equilibrium state
are thebut we assume that they are only scalar ones, such that resp.
available.
four-vectors
onlyis of the form
Hence
$T^{\mu\nu}(\omega_{\beta})$$\omega_{\beta}$$\beta$$u_{\mathrm{t}\mathrm{h}}$$T^{\mu\nu}$(16)$T^{\mu\nu}(\omega_{\beta})=A(\beta^{2}, \ldots)g^{\mu\nu}+B(\beta^{2}, \ldots)u_{\mathrm{t}\mathrm{h}}^{\mu}u_{\mathrm{t}\mathrm{h})}^{\nu}$

and $B$ are scalar functions which depend on the system under consideration.
Combination of equations (13) and (16) yields$A$$e_{\beta}=A(\beta^{2})+B(\beta^{2})$$\mathrm{i}.\mathrm{e}$.

$e_{\beta}$

is a function of$\beta^{2}$

(17)

,

only. The same holds true of$s^{\mu}(\omega_{\beta})=\sigma v_{\mathrm{t}\mathrm{h}}^{\mu};\sigma=s_{\mathrm{e}\mathrm{q}}(e_{\beta})$

As a consequence of equation (17) we can express:$s_{\infty_{1}}(e_{\beta})$, we can write.
$\sigma(\beta^{2})$(18)in terms of

$A$

and

$B$

$\sqrt{\beta^{2}}=\frac{\partial s_{\eta}}{\partial u}(e_{\beta})=\frac{\partial\sigma}{\partial\beta^{2}}(\frac{\partial u_{\beta}}{\partial\beta^{2}})^{-1}=\frac{\sigma’}{A’+B’}$;$\sigma=\int^{\beta^{2}}dx\sqrt{x}(A’(x)+B’(x))=-\int_{\beta^{2}}^{\infty}dx\sqrt{x}(A’(x)+B’(x))$

(19)

.

(20)

; the integration constant is(The prime denotes the derivative w.r.t.
$\tau=0.$)
for
entropyvanishesthechosen so thatis a concave function, and, as a consequence we note that$x\equiv\beta^{2}$$s_{\eta}$$A’+B’<0$ .

(21)

134

With the help of (19) the proof is easy (differentiability assumed, again weand make use of (17) :put$x=\beta^{2}$$)$$0> \frac{\partial^{2}s_{\mathrm{e}\mathrm{q}}}{\partial u^{2}}$

Since

$=$

$\frac{\partial}{\partial u}\frac{\partial s_{\mathrm{e}\mathrm{q}}}{\partial u}(e_{\beta})=\frac{\partial}{\partial x}(\frac{\partial s_{\alpha_{1}}}{\partial u}(e_{\beta}))(\frac{\partial(e_{\beta})}{\partial x})^{-1}$$=$$( \frac{\sigma’’}{e_{\beta}’}-\frac{\sigma’}{e_{\beta}^{;2}}e_{\beta}’’)(e_{\beta}’)^{-1}=\frac{1}{e_{\beta}^{\prime 2}}(\sigma’’-\sigma’\frac{e_{\beta}’’}{e_{\beta}},$$)$, the expression in brackets, which has to be negative, equals
and; differentiating once more, we get$\sigma’=\sqrt{x}e_{\beta}’$$\sigma’’-\sqrt{x}e_{\beta}’’<0$$\sigma’$$\sigma^{n}=\frac{1}{2\sqrt{x}}e_{\beta}’+\sqrt{x}e_{\beta}’’$thus$0> \sigma’’-\sqrt{x}e_{\beta}’’=\frac{1}{2x}\sigma’=A’+B’$

3.3.

Second Law of Thermodynamics

The next task is to give a generalized formulation of the second law of thermodynamics. For systems at rest w.r.t. the given frame of reference, forwhich the energy density is fixed, $e(\omega)=e_{0}$ , the entropy density is maximal. Now consider an arbitrary frame of reference,forwith$u\cdot u=1$ , defining the time axis ofcharacterized by a four-velocity vector
the ffame. For the sake of brevity we shall denote it by “the frame ”. Let
be an arbitrary state in $K$ . Clearly, the energy density ofin the frame is given by$\sqrt{\beta^{2}}=\frac{\partial s_{\mathrm{e}\mathrm{q}}}{\partial u}(e_{0})$$\omega_{\beta}$$u^{\mu},$$u$$\omega=\int_{V^{+}}d\rho(\beta)\omega_{\beta}$$\omega$$u$$e( \omega;u)=u^{\mu}u^{\nu}\omega(T_{\mu\nu})=u^{\mu}u^{\nu}\int_{V^{+}}d\rho(\beta)T_{\mu\nu}(\omega_{\beta})$

.

(22)

Let us keep this quantity fixed, $e(\omega;u)=e_{0}$ , and ask for the state in $K$ which
maximizes the entropy density. The best guess seems to be the following:
Generalized Second Law: The supremum$\sup\{u^{\mu}s_{\mu}(\omega);\omega\in K, e(\omega;u)=e_{0}\}$is attained for the $KMS$ state$\omega_{\beta}$

with$\beta^{\mu}=\tau^{-1}u^{\mu},$.

$\tau^{-1}=\frac{\partial s_{\mathrm{e}\mathrm{q}}}{\partial u}(e_{0})$(23)Due to equations (12) and (18), the supremum is given by$u^{\mu}s_{\mu}(\omega_{\beta})=s_{\mathrm{e}\mathrm{q}}(e_{0})=\sigma(\beta^{2})$

.

(24)

135

Although this cannot be directly verified, it can be shown under theassumption that the following holds:
is maximal if
Restricted Second Law: Given , then.
Together with the Second Law in the rest system, this then implies thefull Generalized Second Law:
withLemma:
is a concave function, andSince, the Restricted Second Law implies the Generalized SecondLaw.
$\beta^{\mu}=$$u^{\mu}s_{\mu}(\omega_{\beta})$$e_{0}$$\sqrt{\beta^{2}}u^{\mu}$$s_{\alpha_{1}}(e)=\sigma(\beta^{2})$$s_{\mathrm{e}\mathrm{q}}(e)$$\sqrt{\beta^{2}}=\partial_{e}s_{\mathrm{e}\mathrm{q}}(e)$In the model of a massless free field, the validity of the Restricted Second
Law can be checked: The energy momentum tensor is is given by$T_{\mu\nu}=:\partial_{\mu}\Phi\partial_{\nu}\Phi$$:- \frac{1}{2}g_{\mu\nu}$:

its expectation value in the equilibrium state$\partial_{\rho}\Phi\partial^{\rho}\Phi:$$\omega_{\beta}$

,

being

$\omega_{\beta}(T_{\mu\nu})=T_{\mu\nu}(\omega_{\beta})=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\int\frac{d^{3}p}{2|p]}\frac{e^{-\beta\overline{p}}}{1-e^{-\beta\overline{p}}}\overline{p}_{\mu}\overline{p}_{\nu}$,is the four-momentum on the mass shell $m=0$ . Calculationwhere, see equations (13) and (17), withofthe help of the above integral, and insertion into (16) yields$\overline{p}=(|p],\vec{p})$$u_{\mathrm{t}\mathrm{h}}^{\mu}(\beta)u_{\mathrm{t}\mathrm{h}}^{\nu}(\beta)T_{\mu\nu}(\omega_{\beta})=A(\beta^{2})+B(\beta^{2})$(25)$T_{\mu\nu}(\omega_{\beta})=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}(\beta^{2})^{-1}(-g_{\mu\nu}+4v_{\mu}^{\mathrm{t}\mathrm{h}}(\beta)u_{\nu}^{\mathrm{t}\mathrm{h}}(\beta))$

Let us denote the constant by , which is positive. Evidently, we have$c$$A(x)=-cx^{-2}$ ,$B(x)=4cx^{-2}$(The relation $B(x)=-4A(x)$ holds due to (16) whenever(20) it then follows that$\sigma(x)=\int_{x}^{\infty}y^{1/2}6cx^{-3}dy=4cx^{-3/2}$$T_{\mu}^{\mu}=0.$

) Rom

,

$\Sigma’(x)=-cx^{-5/2}(\frac{e_{0}}{c}x^{2}+1)^{-1/2}(\frac{e_{0}}{c}x^{2}+3)$.

Since $\Sigma’<0$ , the maximum is reached at the upper limit of the range of ,. at $\xi=1$ , as assumed above.
$x$$\mathrm{i}.\mathrm{e}$136

References[1] R. Haag and I. Ojima: Ann. Inst. Henri Poincar\’e, 64(4), 385, 1996.

[2] H. Bostelmann: Zustandskeime in der lokalen Quantenfeldtheorie.
Diploma Thesis, Institit f\"ur Theoretische Physik der Universit\"at
G\"ottingen, May 1998.
[3] R.Brunetti, K.Redenhagen and M.K\"ohler:

633, 1996.

Comm. Math. Phys, 180,[4] J. Bros and D. Buchholz: Nucl. Phys., B429, 291, 1994.
[5] I. Ojima: Lett. Math. Phys., 11, 73, 1986.