Finite Temperature Field Theories: Review
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Abstract
In this section, we briefly summarize our basic knowledge of quantum field theories, required for the investigation of equilibrium (and eventually nonequilibrium) features of quantum systems with a very large (infinite) number of degrees of freedom. The best representation of the dynamical variables are fields with a Lagrangian describing local interactions among them. The thus defined field theory can then be quantized and eventually put in a heat bath.
Keywords
Spectral Function Ward Identity External Current Path Integral Representation Retarded PropagatorThis introductory section very concisely summarizes the most important features of the process of constructing quantum field theories at zero and nonzero temperatures. There are excellent books and monographs in the literature where the interested reader can find further details. Without trying to be complete, we give here some basic references. For general questions on quantum field theory, one can make use of several books [1, 2, 3]. For the more specialized problems of finitetemperature field theory, a very useful review can be found in [4], and two more recent books are also available [5] and [6]. It is often useful to think about a field theory as the continuum limit of a theory defined on a spacetime lattice: an excellent book on this topic was written by Montvay and Münster [7]. Our review of renormalization in the next chapter is largely based on the book of Collins [8]. Beyond these basic works, there are numerous excellent books and papers on more specific subjects that we are going to quote in the text.
Throughout these notes, field variables are defined in spacetime points x of flat ddimensional Minkowskian or Euclidean spacetime M isomorphic to R^{ d }. It is often useful to have an observer splitting the spacetime into time and space extensions, denoted by x = (t, x); here x is a (d − 1)dimensional vector.
2.1 Review of Classical Field Theory
In classical field theory, the basic object is the field A: M → V, where V is some vector space. The state of the system at time t is characterized by the configuration A(t, x). The dynamics of the system is determined by the action S, which is a realvalued functional of the field configurations: S: A ↦ R. The classically realizable time evolutions are solutions of the equation of motion (EoM) manifesting themselves as the extrema of the action δ S∕δ A = 0; these solutions are also called physical configurations. In local field theories, the action can be written as an integral over the Lagrangian density, \(S =\int d^{d}x\,\mathcal{L}\). In the fundamental field theories, the Lagrangian density depends only on the field and its first derivatives: \(\mathcal{L}(A(x),\partial _{\mu }A(x))\). In this case, we can introduce the canonically conjugate field \(\varPi = \partial \mathcal{L}/\partial \dot{A}\).
We can apply to the configurations transformations represented by bijections U: A ↦ A′. The transformations of the system form a group. We will consider only pointwise transformations, where the transformed new field at a given position depends on the original field at another position, i.e., A′(x′) = U_{ V }(A(x)), where x′ = U_{ M }(x). The transformation is called internal if U_{ M } = 1 (identity); otherwise, it is an external transformation. We often encounter transformations determined by continuously varying parameters, i.e., when a certain subgroup of all transformations forms a Lie group. In this case, the transformation U can be characterized by a set of parameters c ∈ R^{ n }. In the standard representation, the transformation is written as \(U(c) = e^{ic_{a}T_{a}}\), where T_{ a } are the generators of the transformation.
The transformation is a symmetry if S[A′] = S[A]. This also means that if A is an extremum of S (i.e., realizable motion), then A′ is an extremum, too. In several cases, we expect that the system exhibits some abstractly defined symmetry group. In this case, the fields are classified as irreducible representations of the abstract group, and S is built up from the group invariants constructed from the field products.
2.2 Quantization and Path Integral
The main difference between a classical and a quantum system is the characterization of each of their states. While in a classical system the state could be fully described by the configuration of the generalized coordinates and the canonically conjugated momenta, in the quantum system we introduce a separate Hilbert space H for the states. The physical states are Ψ[A] ∈ H of unit norm  Ψ[A]  = 1.
The transformations of the quantum system form U: H → H isomorphisms. These are (anti)linear maps that map a state onto a state, i.e., it conserves the norm; therefore, the transformation must be (anti)unitary, \(U^{\dag } = U^{1}\). In the continuous case, \(U_{c} = e^{ic_{a}T_{a}}\), just as in the classical case; if U is unitary, then the generator T is Hermitian, \(T^{\dag } = T\).
The generator of a transformation does not change under the effect of the same transformation T′ = U^{†}TU = T. It makes it possible to identify the generator of a symmetry transformation as the conserved quantity belonging to this transformation: in field theory, this is the conserved quantity coming from Noether’s theorem. Therefore, the generators of the space and time translations are the Noether charges coming from the energy–momentum tensor, i.e., the conserved momentum and energy, respectively. After quantization, these become the momentum operator \(\hat{P}_{i}\) and Hamilton operator \(\hat{H}\), and so the space and time translations are \(e^{i\hat{P}_{i}x^{i}/\hslash }\) and \(e^{i\hat{H}t/\hslash }\), respectively (\(\hslash \) is introduced in order to have dimensionless quantities in the exponent).
The infinitesimal transformations of the system are interpreted as measurements. In an eigenstate, we obtain a definite value for the result of the measured quantity; in other cases, the projection of the arising state onto the eigenstates gives the probability amplitude of the possible outcomes of the measurement. Two measurements are not interchangeable if the corresponding generators do not commute.
The eigenvectors of \(\hat{A}\) (and also those of \(\hat{\varPi }\)) form an orthonormal basis. In this basis, we can expand any state, and using the projectors on the eigenstates also any operator. In particular, we can expand the time translation operator \(e^{i\hat{H}t}\).
In the case of fermions, the situation is more complicated, because the field measurement operators \(\hat{\varPsi }(t,\mathbf{x})\) do not commute at a given time. To resolve the problem, one first introduces the Grassmann algebra \(\mathcal{G}\). It is a (graded) complex algebra generated by the unity and elements e_{ i } for which the algebraic product is antisymmetric \(e_{i}e_{j} = e_{j}e_{i}\). For the physical applications, one also needs a star operation that connects the generator elements into pairs ∗: e → e^{∗}. We can define a Hilbert space over the Grassmann algebra \(H(\mathcal{G})\), which means that if \(\left \vert \psi _{1}\right \rangle,\left \vert \psi _{2}\right \rangle \in H(\mathcal{G})\) and \(g_{1},g_{2} \in \mathcal{G}\), then \(g_{1}\left \vert \psi _{1}\right \rangle + g_{2}\left \vert \psi _{2}\right \rangle \in H(\mathcal{G})\). The scalar product is Grassmannalgebravalued, \(\left \langle \psi _{1}\vert \psi _{2}\right \rangle \in \mathcal{G}\). In the physical applications, one introduces two Grassmann algebra generators for each spatial position and for each fermionic component \(e_{\alpha,\mathbf{x}}\) and e_{α, x}^{∗}. The field operators \(\hat{\varPsi }(t,\mathbf{x})\) and \(\hat{\varPsi }^{\dag }(t,\mathbf{x})\) act linearly on \(H(\mathcal{G})\), i.e., \(\hat{\varPsi }(t,\mathbf{x})(g_{1}\left \vert \psi _{1}\right \rangle + g_{2}\left \vert \psi _{2}\right \rangle ) = g_{1}\hat{\varPsi }(t,\mathbf{x})\left \vert \psi _{1}\right \rangle + g_{2}\hat{\varPsi }(t,\mathbf{x})\left \vert \psi _{2}\right \rangle\), and similarly for \(\hat{\varPsi }^{\dag }(t,\mathbf{x})\).
The case of a single degree of freedom discussed above can be taken over for more fermionic dynamical variables without any problem. The most important finding is the integral formula representing the spectral decomposition of the identity and the trace formula.
In real time, however, if the time arguments appear without any definite order, the contour runs back and forth several times. If for a certain section it is monotonic, those subsections can be merged into a common section; e.g., if \(t_{1} < t_{2} < t_{3}\), then we can write instead of \(t_{1} \rightarrow t_{2} \rightarrow t_{3}\) simply \(t_{1} \rightarrow t_{3}\). Moreover, we can always plug in a unit operator as \(e^{iH(t't)}e^{iH(t't)}\), i.e., we can include a bypass anywhere in the time chain: instead of \(t_{n} \rightarrow t_{n+1}\), we can write \(t_{n} \rightarrow t' \rightarrow t_{n+1}\). According to these observations, the time contour can be standardized as \(t_{i} \rightarrow t_{f} \rightarrow t_{i} \rightarrow \cdots \rightarrow t_{f} \rightarrow t_{i}\), where a sufficient number of backandforth sections must be allowed for it to contain (taking into account the ordering) all the operators. This form depends on the operators only through their number. On the sections where the time flows as t_{ i } → t_{ f }, the contour ordering is time ordering, while along the backwardrunning contours, the contour ordering is antitimeordering.
2.3 Equilibrium
The consequence of equilibrium is that we have time translationinvariance in the observables. Therefore, we can put the initial density matrix anywhere in time, even \(t_{i} \rightarrow \infty \). But in this case, all propagation between contour sections C_{1, 2} and C_{3} is suppressed, since we would need infinitely long propagation. This means that segment 3 factorizes from segments 1 and 2. The formalism that uses exclusively segment 3 is called Euclidean, or imaginary, time or the Matsubara formalism. The theory built on contours 1 and 2 is called real time, or the Keldysh formalism.
2.4 Propagators
2.5 Free Theories, Propagators, Free Energy
These expressions are finite unless α = 1 and  μ  > m (the \(\vert \mu \vert \rightarrow m + 0\) case is still finite). In this pathological case, there appears a pole at \(p = \sqrt{\mu ^{2 }  m^{2}}\). Physically, this phenomenon is connected to the Bose–Einstein condensation: in response to trying to increase the number density of particles, the system generates a finite field condensate.
 If T ≪ m and μ = 0, one has, for both the fermionic and bosonic cases,$$\displaystyle{ p = T\left (\frac{mT} {2\pi } \right )^{3/2}e^{\beta m},\qquad \frac{\varepsilon } {p} = \frac{3} {2} + \frac{m} {T}. }$$(2.89)
 If T ≫ m, μ, one has$$\displaystyle{ \frac{p_{b}} {T^{4}} = \frac{\pi ^{2}} {90},\qquad p_{f} = \frac{7} {8}p_{b},\qquad \varepsilon = 3p. }$$(2.90)
2.6 Perturbation Theory
If the action contains nonquadratic terms, then we must rely on some approximations for the evaluation of the path integral. Under the assumption that the quadratic (Gaussian) approximation captures the main physical features of the system, only slight nonlinear modifications are expected. Then one can use perturbation theory.

draw a graph with points and links (lines);

associate the points where only a single link ends (external legs) to the fields \(A_{i_{k}}(p_{k})\), \(k = 1\mathop{\ldots },n\), where n is the number of this type of point;

associate the points with v links to the piece in the interaction S_{ int } containing the product of v field operators. These points are called interaction vertices;

associate propagators to the links connecting any two points. A number of externally determined propagators enter into each vertex and generate virtual fields propagating along the internal lines of the diagram.

external legs force the joining propagator to have momentum p_{ k };

a link connecting points with indices i and j is represented by iG^{(ij)}(q);

a vertex having incoming momenta \(q_{1},\mathop{\ldots },q_{v}\) yields a contribution \(ig(2\pi )^{d}\delta (\sum _{a=1}^{v}q_{a})\).
In the last step, one multiplies all these contributions, and an extra factor 1∕m! is included in the mth order of the perturbation theory. Finally, one evaluates the integrals over the momenta flowing through the internal links: \(\varPi _{i}\int \!\frac{d^{d}q_{ i}} {(2\pi )^{d}} \,\).
2.7 Functional Methods
Perturbation theory can be made more efficient if we realize that the only quantities that need calculation are the loop integrals arising from the rules for setting up Feynman diagrams described in the last section. Therefore, it is worth developing a formalism that concentrates solely on them.
The value of Γ thus defined has many interesting properties. Diagrammatically, \(\left \langle A_{x}\right \rangle _{J}\) is a onepoint function. Fixing the value of \(\bar{A}\) means fixing the value of the onepoint function. Therefore, we must ensure the vanishing of the perturbative corrections to the onepoint function. This requirement is equivalent to the omission of all diagrams that contain a part connected to the rest by a single line. Such diagrams are called oneparticle reducible diagrams. Those diagrams that remain connected after cutting a line form the class of oneparticle irreducible (1PI) diagrams. Therefore, Γ can be interpreted as being the generator of the 1PI diagrams.
2.7.1 TwoPoint Functions and SelfEnergies
Although it looks quite simple, (2.112) in fact avoids by resummation a series of individually increasingly divergent contributions. Namely, the free propagator exhibits a pole at p^{2} = m^{2} on the mass shell. Near the mass shell, we have \(p^{2} = m^{2} + x\), and the free propagator behaves as \(G_{0} \sim 1/x\). Therefore, the nth term in the DS series is proportional to 1∕x^{ n }. These terms are therefore more and more divergent at a finite value of the momentum: this is characteristic of an infrared (IR) divergence. The DS series resums the most relevant terms and provides the propagator G free of unphysical divergences. It also may have a pole, but that is already physical: no further diagrams can change it. This logic will be followed later in performing the resummation of IR divergent (IR sensitive) series.
2.7.2 Higher nPoint Functions
These formulas correspond to the treelevel relations between the npoint functions and the vertices when vertex strengths are derived from the classical action. The derivatives of the effective action therefore play the role of the classical vertices: hence the name proper vertices. In contrast to the classical theory, quantum fluctuations produce nonzero values for arbitrary high npoint functions.
2.8 The TwoParticle Irreducible Formalism
In the previous subsection we investigated the generator of the 1PI diagrams, where the system is forced by a choice of appropriate external current distribution to take a predetermined value \(\bar{A} = \left \langle A\right \rangle\) for the expectation value of the field operator. We can go on with this logic and try to impose other constraints on the path integral. The next step is to fix the value of the twopoint functions as well. Technically, what we have to do is to assign an external current to the twopoint function and determine its value from the requirement that the exact twopoint function coincide with the a priori chosen expression.
In special cases, we can determine \(\varGamma [\bar{A},\bar{G}]\). If there is no quantum correction at all, then the path integral substitutes the solution of the classical EoM, which is the same as the first equation of (2.129) with Γ → S. Classically, \(\bar{G} = 0\), and so the classical 2PI effective action reverts to the classical 1PI case, i.e., Γ = S.
A peculiarity of the 2PI formalism is that we can access the same correlation functions in different ways. For example, the selfenergy can be expressed from (2.137) as the derivative of the 2PI action with respect to \(\bar{G}\), but it is also the second derivative with respect to \(\bar{A}\). This ambiguity remains true for all higher point functions. For the exact correlation functions, these definitions yield the same result, but in the perturbative expansion, usually we find deviations. The derivative of the 1PI effective action, for example, always respects the symmetry of the Lagrangian, and so it satisfies Goldstone’s theorem in the case of spontaneous symmetrybreaking (SSB) of a continuous symmetry. This is not true for the solution of the 2PI equation (2.132) if we substitute back the physical value of the background. The reason is that Goldstone’s theorem is satisfied at each order of perturbation theory as a result of subtle cancellations between the selfenergy and the vertex corrections. Therefore, when we resum all the selfenergy diagrams up to a certain order using the 2PI formalism and leave the vertex corrections at their treelevel perturbative value, we easily find a mismatch. We will return to this point later, in Sect. 4.7
We can also perform a Legendre transformation to obtain the free energy Γ[V_{ k }], which is the functional of V_{ k }, where we can also express the external sources R^{(k)} through V_{ k }. The physical value of the vertices comes from the requirements \(\delta \varGamma [V _{k}]/\delta V _{j} = 0\). Since we fixed all the correlation functions up to the npoint functions, the perturbative expansion should contain only nparticleirreducible (nPI) diagrams, which remain connected even after cutting n internal lines.
2.9 Transformations of the Path Integral
2.9.1 Equation of Motion, Dyson–Schwinger Equations
2.9.2 Ward Identities from a Global Symmetry
 1.If we integrate over the variable i (which also contains the spacetime integration), we obtainThe righthand side is the total change in the expectation value$$\displaystyle{ 0 =\sum _{ k=1}^{n}\left \langle A_{ a1}\ldots A_{a_{k1}}\varDelta A_{a_{k}}A_{a_{k+1}}\ldots A_{a_{n}}\right \rangle. }$$(2.156)This means that the expectation value of every npoint function is invariant under the global symmetry transformation. Note that this is true for the complete npoint function, not just the connected part.$$\displaystyle{ 0 =\varDelta \left \langle A_{a_{1}}\ldots A_{a_{n}}\right \rangle. }$$(2.157)
 2.In the case of linear continuous transformations, \(\varDelta A_{i} = T_{ij}A_{j}\). Taking the integrated form of the effective action expression with a constant (i.e., zeromomentum) A field, we havewhere one sums over repeated indices. Taking its functional derivative yields$$\displaystyle{ 0 = \frac{\partial \varGamma } {\partial A_{i}}T_{ij}A_{j}, }$$(2.158)In the physical point, \(\frac{\partial \varGamma } {\partial A_{i}} = 0\), but the field itself can take a (constant) expectation value in the SSB case. Therefore,$$\displaystyle{ \frac{\partial ^{2}\varGamma } {\partial A_{k}\partial A_{i}}T_{ij}A_{j} + \frac{\partial \varGamma } {\partial A_{i}}T_{ik} = 0. }$$(2.159)which is Goldstone’s theorem: the inverse propagator has a zero mode at zero momentum in the SSB case.$$\displaystyle{ \frac{\partial ^{2}\varGamma } {\partial A_{k}\partial A_{i}}T_{ij}A_{j} = 0, }$$(2.160)
 3.Remaining in the class of linearly represented symmetries, we write down the Ward identity for two fields explicitly, displaying the spacetime indices:We assume that we are in the symmetric case (the SSB case is only formally more complicated). One introduces for the lefthand side a 1PI proper vertex:$$\displaystyle{ \partial _{x}^{\mu }\left \langle j_{\mu }(x)A_{ i}(y)A_{j}(z)\right \rangle = i\delta (x  y)T_{ik}\left \langle A_{k}(y)A_{j}(z)\right \rangle + i\delta (x  z)T_{jk}\left \langle A_{i}(y)A_{k}(z)\right \rangle. }$$(2.161)The Ward identity takes the following form after passage to Fourier space:$$\displaystyle{ \left \langle j_{\mu }(x)A_{i}(y)A_{j}(z)\right \rangle = iG_{ii'}(y  y')\,iG_{jj'}(z  z')\,\varGamma _{i'j'}^{\mu }(x,y',z'). }$$(2.162)where k + p + q = 0. Multiplying by the inverse propagators gives us$$\displaystyle{ G_{ii'}(p)G_{jj'}(q)(ik_{\mu })\varGamma _{i'j'}^{\mu }(k,p,q) = T_{ ik}G_{jk}(q) + T_{jk}G_{ik}(p), }$$(2.163)There is a special case, in which the propagation is diagonal (\(G_{ij} = G_{i}\delta _{ij}\) and \(T_{ij} = T_{ji}\). Then$$\displaystyle{ (ik_{\mu })\varGamma _{ij}^{\mu }(k,p,q) = G_{ ik}^{1}(p)T_{ kj} + G_{jk}^{1}(q)T_{ ki}. }$$(2.164)This is an often used consequence of the Ward identity.$$\displaystyle{ (ik_{\mu })\varGamma _{ij}^{\mu }(k,p,q) = T_{ ij}\left (G_{i}^{1}(p)  G_{ j}^{1}(q)\right ). }$$(2.165)
2.10 Example: Φ^{4} Theory at Finite Temperature
Footnotes
 1.
We remark that in (2 + 1)dimensional field theories, there is a possibility to define particles with any α with unit length. These are called anyons.
 2.
This vacuum contribution used to be attributed to the Casimir effect, where we measure the energy difference arising when the volume of the quantization space is changed. It is also worth noting that fermions and bosons contribute with opposite signs, which means that in supersymmetric models, the net zeropoint contribution is zero.
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