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March 2024
Basics of Thermal Field Theory
A Tutorial on Perturbative Computations 1
Mikko Lainea^ and Aleksi Vuorinenb
aAEC, Institute for Theoretical Physics, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland bDepartment of Physics, University of Helsinki, P.O. Box 64, FI-00014 University of Helsinki, Finland
Abstract
These lecture notes, suitable for a two-semester introductory course or self-study, offer an elemen- tary and self-contained exposition of the basic tools and concepts that are encountered in practical computations in perturbative thermal field theory. Selected applications to heavy ion collision physics and cosmology are outlined in the last chapter.
(^1) An earlier version of these notes is available as an ebook (Springer Lecture Notes in Physics 925) at dx.doi.org/10.1007/978-3-319-31933-9; an eprint can be found at arxiv.org/abs/1701.01554; the very latest version is kept up to date at www.laine.itp.unibe.ch/basics.pdf.
- 1 Quantum mechanics General outline iii
- 1.1 Path integral representation of the partition function
- 1.2 Evaluation of the path integral for the harmonic oscillator
- 2 Free scalar fields
- 2.1 Path integral for the partition function
- 2.2 Evaluation of thermal sums and their low-temperature limit
- 2.3 High-temperature expansion
- 3 Interacting scalar fields
- 3.1 Principles of the weak-coupling expansion
- 3.2 Problems of the naive weak-coupling expansion
- 3.3 Proper free energy density to O(λ): ultraviolet renormalization
- 4 Fermions
- 4.1 Path integral for the partition function of a fermionic oscillator
- 4.2 The Dirac field at finite temperature
- 5 Gauge fields
- 5.1 Path integral for the partition function
- 5.2 Weak-coupling expansion
- 5.3 Thermal gluon mass
- 5.4 Free energy density to O(g^3 )
- 6 Low-energy effective field theories
- 6.1 The infrared problem of thermal field theory
- 6.2 Dimensionally reduced effective field theory for hot QCD
- 7 Finite density
- 7.1 Complex scalar field and effective potential
- 7.2 Dirac fermion with a finite chemical potential
- 8 Real-time observables
- 8.1 Different Green’s functions
- 8.2 From a Euclidean correlator to a spectral function
- 8.3 Real-time formalism
- 8.4 Hard Thermal Loops
- 9 Applications
- 9.1 Thermal phase transitions
- 9.2 Bubble nucleation rate
- 9.3 Particle production rate
- 9.4 Embedding rates in cosmology
- 9.5 Evolution of a long-wavelength field in a thermal environment
- 9.6 Linear response theory and transport coefficients
- 9.7 Equilibration rates / damping coefficients
- 9.8 Resonances in medium
- Appendix: Extended Standard Model in Euclidean spacetime
- Index
Foreword
These notes are based on lectures delivered at the Universities of Bielefeld and Helsinki, between 2004 and 2015, as well as at a number of summer and winter schools, between 1996 and 2018. The early sections were strongly influenced by lectures by Keijo Kajantie at the University of Helsinki, in the early 1990s. Obviously, the lectures additionally owe an enormous gratitude to existing text books and literature, particularly the classic monograph by Joseph Kapusta.
There are several good text books on finite-temperature field theory, and no attempt is made here to join that group. Rather, the goal is to offer an elementary exposition of the basics of the field, in an explicit “hands-on” style which can hopefully more or less directly be transported to the classroom. The presentation is meant to be self-contained and display also intermediate steps. The idea is, roughly, that each numbered section could constitute a single lecture. Referencing is sparse; on more advanced topics, as well as on historically accurate references, the reader is advised to consult the text books and review articles in refs. [0.1]–[0.17].
These notes could not have been put together without the helpful influence of many people, varying from students with persistent requests for clarification; colleagues who have used parts of an early version of these notes in their own lectures and shared their experiences with us; colleagues whose interest in specific topics has inspired us to add corresponding material to these notes; alert readers who have informed us about typographic errors and suggested improvements; and collaborators from whom we have learned parts of the material presented here. Let us gratefully acknowledge in particular Gert Aarts, Chris Korthals Altes, Dietrich B¨odeker, Yannis Burnier, Stefano Capitani, Simon Caron-Huot, Jacopo Ghiglieri, Ioan Ghisoiu, Kimmo Kainulainen, Keijo Kajantie, Aleksi Kurkela, Harvey Meyer, Guy Moore, Paul Romatschke, Kari Rummukainen, York Schr¨oder, Mikhail Shaposhnikov, Markus Thoma, Tanmay Vachaspati, and Mikko Veps¨al¨ainen.
Mikko Laine and Aleksi Vuorinen
i
General outline
Physics context
From the physics point of view, there are two important contexts in which relativistic thermal field theory is being widely applied: cosmology and the theoretical description of heavy ion collision experiments.
In cosmology, the temperatures considered vary hugely, ranging from T ≃ 1015 GeV to T ≃ 10 −^3 eV. Contemporary challenges in the field include figuring out explanations for the existence of dark matter, the observed antisymmetry in the amounts of matter and antimatter, and the formation of large-scale structures from small initial density perturbations. (The origin of initial density perturbations itself is generally considered to be a non-thermal problem, associated with an early period of inflation.) An important further issue is that of equilibration, i.e. details of the processes through which the inflationary state turned into a thermal plasma, and in particular what the highest temperature reached during this epoch was. It is notable that most of these topics are assumed to be associated with weak or even superweak interactions, whereas strong interactions (QCD) only play a background role. A notable exception to this is light element nucleosynthesis, but this well-studied topic is not in the center of our current focus.
In heavy ion collisions, in contrast, strong interactions do play a major role. The lifetime of the thermal fireball created in such a collision is ∼ 10 fm/c and the maximal temperature reached is in the range of a few hundred MeV. Weak interactions are too slow to take place within the lifetime of the system. Prominent observables are the yields of different particle species, the quenching of energetic jets, and the hydrodynamic properties of the plasma that can be deduced from the observed particle yields. An important issue is again how fast an initial quantum-mechanical state turns into an essentially incoherent thermal plasma.
Despite many differences in the physics questions posed and in the microscopic forces underly- ing cosmology and heavy ion collision phenomena, there are also similarities. Most importantly, gauge interactions (whether weak or strong) are essential in both contexts. Because of asymptotic freedom, the strong interactions of QCD also become “weak” at sufficiently high temperatures. It is for this reason that many techniques, such as the resummations that are needed for developing a formally consistent weak-coupling expansion, can be applied in both contexts. The topics covered in the present notes have been chosen with both fields of application in mind.
Organization of these notes
The notes start with the definition and computation of basic “static” thermodynamic quantities, such as the partition function and free energy density, in various settings. Considered are in turn quantum mechanics (sec. 1), free and interacting scalar field theories (secs. 2 and 3, respectively), fermionic systems (sec. 4), and gauge fields (sec. 5). The main points of these sections include the introduction of the so-called imaginary-time formalism; the functioning of renormalization at finite temperature; and the issue of infrared problems that complicates almost every computation in relativistic thermal field theory. The last of these issues leads us to introduce the concept of effective field theories (sec. 6), after which we consider the changes caused by the introduction of a finite density or chemical potential (sec. 7). After these topics, we move on to a new set of observ-
iii
ables, so-called real-time quantities, which play an essential role in many modern phenomenological applications of thermal field theory (sec. 8). In the final chapter of the book, a number of concrete applications of the techniques introduced are discussed (sec. 9).
We note that secs. 1–7 are presented on an elementary and self-contained level and require no background knowledge beyond statistical physics, quantum mechanics, and rudiments of quantum field theory. They could constitute the contents of a one-semester basic introduction to perturbative thermal field theory. In sec. 8, the level increases gradually, and parts of the discussion in sec. 9 are already close to the research level, requiring more background knowledge. Conceivably the topics of secs. 8 and 9 could be covered in an advanced course on perturbative thermal field theory, or in a graduate student seminar. In addition the whole book is suitable for self-study, and is then advised to be read in the order in which the material has been presented.
Recommended literature
A pedagogical presentation of thermal field theory, concentrating mostly on Euclidean observables and the imaginary-time formalism, can be found in ref. [0.1]. The current notes borrow significantly from this classic treatise.
In thermal field theory, the community is somewhat divided between those who find the imaginary- time formalism more practicable, and those who prefer to use the so-called real-time formalism from the beginning. Particularly for the latter community, the standard reference is ref. [0.2], which also contains an introduction to particle production rate computations.
A modern textbook, partly an update of ref. [0.1] but including also a full account of real-time observables, as well as reviews on many recent developments, is provided by ref. [0.3].
Lecture notes on transport coefficients, infrared resummations, and non-equilibrium phenomena such as thermalization, can be found in ref. [0.4]. Reviews with varying foci are offered by refs. [0.5]– [0.16].
Finally, an extensive review of efforts to approach a non-perturbative understanding of real-time thermal field theory has been presented in ref. [0.17].
iv
1. Quantum mechanics
Abstract: After recalling some basic concepts of statistical physics and quantum mechanics, the partition function of a harmonic oscillator is defined and evaluated in the standard canonical for- malism. An imaginary-time path integral representation is subsequently developed for the partition function, the path integral is evaluated in momentum space, and the earlier result is reproduced upon a careful treatment of the zero-mode contribution. Finally, the concept of 2-point functions (propagators) is introduced, and some of their key properties are derived in imaginary time.
Keywords: Partition function, Euclidean path integral, imaginary-time formalism, Matsubara modes, 2-point function.
1.1. Path integral representation of the partition function
Basic structure
The properties of a quantum-mechanical system are defined by its Hamiltonian, which for non- relativistic spin-0 particles in one dimension takes the form
Hˆ = pˆ
2 2 m
where m is the particle mass. The dynamics of the states |ψ〉 is governed by the Schr¨odinger equation,
iℏ
∂t |ψ〉 = Hˆ|ψ〉 , (1.2)
which can formally be solved in terms of a time-evolution operator Uˆ (t; t 0 ). This operator satisfies the relation |ψ(t)〉 = Uˆ(t; t 0 )|ψ(t 0 )〉 , (1.3)
and for a time-independent Hamiltonian takes the explicit form
Uˆ (t; t 0 ) = e−^ ℏi^ Hˆ(t−t^0 )^. (1.4)
It is useful to note that in the classical limit, the system of eq. (1.1) can be described by the Lagrangian
L = LM =
m x˙^2 − V (x) , (1.5)
which is related to the classical version of the Hamiltonian via a simple Legendre transform:
p ≡
∂LM
∂ x˙ , H = ˙xp − LM = p^2 2 m
Returning to the quantum-mechanical setting, various bases can be chosen for the state vectors. The so-called |x〉-basis satisfies the relations
〈x|xˆ|x′〉 = x〈x|x′〉 = x δ(x − x′) , 〈x|ˆp|x′〉 = −iℏ ∂x〈x|x′〉 = −iℏ ∂x δ(x − x′) , (1.7)
whereas in the energy basis we simply have
Hˆ|n〉 = ǫn|n〉. (1.8)
An important concrete realization of a quantum-mechanical system is provided by the harmonic oscillator, defined by the potential
V (ˆx) ≡
mω^2 xˆ^2. (1.9)
In this case the energy eigenstates |n〉 can be found explicitly, with the corresponding eigenvalues equalling
ǫn = ℏω
n +
, n = 0, 1 , 2 ,.... (1.10)
All the states are non-degenerate.
It turns out to be useful to view (quantum) mechanics formally as (1+0)-dimensional (quantum) field theory: the operator ˆx can be viewed as a field operator φˆ at a certain point, implying the correspondence xˆ ↔ φˆ( 0 ). (1.11)
In quantum field theory operators are usually represented in the Heisenberg picture; correspond- ingly, we then have xˆH (t) ↔ φˆH (t, 0 ). (1.12)
In the following we adopt an implicit notation whereby showing the time coordinate t as an argument of a field automatically implies the use of the Heisenberg picture, and the corresponding subscript is left out.
Canonical partition function
Taking our quantum-mechanical system to a finite temperature T , the fundamental quantity of interest is the partition function, Z. We employ the canonical ensemble, whereby Z is a function of T ; introducing units in which kB = 1 (i.e., There ≡ kBTSI-units), the partition function is defined by
Z(T ) ≡ Tr [e−β^ Hˆ^ ] , β ≡ 1 T
where the trace is taken over the full Hilbert space. From this quantity, other observables, such as the free energy F , entropy S, and average energy E can be obtained via standard relations:
F = −T ln Z , (1.14) S = −
∂F
∂T = ln^ Z^ +^
T Z Tr [ Heˆ −β^ Hˆ^ ] = − F T +^
E
T ,^ (1.15)
E =
Z Tr [ Heˆ −β^ Hˆ^ ]. (1.16)
Let us now explicitly compute these quantities for the harmonic oscillator. This becomes a trivial exercise in the energy basis, given that we can immediately write
Z =
∑^ ∞
n=
〈n|e−β^ Hˆ^ |n〉 =
∑^ ∞
n=
e−βℏω(^ (^12) +n) = e−βℏω/^2 1 − e−βℏω^
2 sinh
( (^) ℏω 2 T
Consequently,
F = T ln
e ℏ 2 ωT − e−^ ℏ 2 ωT^ ) = ℏω 2
1 − e−βℏω
A crucial trick at this point is to insert
1 ˆ =
dpi 2 πℏ |pi〉〈pi| , i = 1,... , N , (1.28)
on the left side of each exponential, with i increasing from right to left; and
ˆ 1 =
dxi |xi〉〈xi| , i = 1,... , N , (1.29)
on the right side of each exponential, with again i increasing from right to left. Thereby we are left to consider matrix elements of the type
〈xi+1 |pi〉〈pi|e−^ ǫ^ ℏ^ Hˆ( ˆp,ˆx)|xi〉 = e
ipixi+ ℏ (^) 〈pi|e−^ ǫ ℏ H(pi^ ,xi)+O(ǫ^2 )|xi〉
= exp
ǫ ℏ
[
p^2 i 2 m − ipi
xi+1 − xi ǫ
]}
Moreover, we note that at the very right, we have
〈x 1 |x〉 = δ(x 1 − x) , (1.31)
which allows us to carry out the integral over x. Similarly, at the very left, the role of 〈xi+1 | is played by the state 〈x| = 〈x 1 |. Finally, we remark that the O(ǫ) correction in eq. (1.30) can be eliminated by sending N → ∞.
In total, we can thus write the partition function in the form
Z = lim N →∞
∫ [^ ∏N
i=
dxidpi 2 πℏ
]
exp
∑^ N
j=
ǫ
[ (^) p 2 j 2 m − ipj
xj+1 − xj ǫ
]}∣∣∣
∣x N +1 ≡^ x^1 , ǫ^ ≡^ βℏ/N
which is often symbolically expressed as a “continuum” path integral
Z =
x(βℏ)=x(0)
Dx D
p 2 πℏ
exp
∫ (^) βℏ
0
dτ
[
[p(τ )]^2 2 m − ip(τ ) ˙x(τ ) + V (x(τ ))
]}
The integration measure here is understood as the limit indicated in eq. (1.32); the discrete xi’s have been collected into a function x(τ ); and the maximal value of the τ -coordinate has been obtained from ǫN = βℏ.
Returning to the discrete form of the path integral, we note that the integral over the momenta pi is Gaussian, and can thereby be carried out explicitly: ∫ (^) ∞
−∞
dpi 2 πℏ exp
ǫ ℏ
[
p^2 i 2 m − ipi
xi+1 − xi ǫ
]}
m 2 πℏ ǫ exp
[
m(xi+1 − xi)^2 2 ℏ ǫ
]
Using this, eq. (1.32) becomes
Z = lim N →∞
∫ [^ ∏N
i=
√^ dxi 2 πℏ ǫ/m
]
exp
∑^ N
j=
ǫ
[
m 2
xj+1 − xj ǫ
]}∣∣∣
∣x N +1 ≡^ x^1 , ǫ^ ≡^ βℏ/N
which may also be written in a continuum form. Of course the measure then contains a factor which appears quite divergent at large N ,
C ≡
m 2 πℏ ǫ
)N/ 2
= exp
[
N
ln
mN 2 πℏ^2 β
)]
This factor is, however, independent of the properties of the potential V (xj ) and thereby contains no dynamical information, so that we do not need to worry too much about the apparent divergence. For the moment, then, we can simply write down a continuum “functional integral”,
Z = C
x(βℏ)=x(0)
Dx exp
∫ (^) βℏ
0
dτ
[
m 2
dx(τ ) dτ
]}
Let us end by giving an “interpretation” to the result in eq. (1.37). We recall that the usual quantum-mechanical path integral at zero temperature contains the exponential
exp
i ℏ
dt LM
, LM =
m 2
dx dt
− V (x). (1.38)
We note that eq. (1.37) can be obtained from its zero-temperature counterpart with the following recipe [1.1]:
(i) Carry out a Wick rotation, denoting τ ≡ it.
(ii) Introduce LE ≡ −LM (τ = it) = m 2
dx dτ
(iii) Restrict τ to the interval (0, βℏ). (iv) Require periodicity of x(τ ), i.e. x(βℏ) = x(0).
With these steps (and noting that idt = dτ ), the exponential becomes
exp
i ℏ
dt LM
(i) −→−(iv) exp
SE
≡ exp
∫ (^) βℏ
0
dτ LE
where the subscript E stands for “Euclidean”. Because of step (i), the path integral in eq. (1.40) is also known as the imaginary-time formalism. It turns out that this recipe works, with few modifications, also in quantum field theory, and even for spin-1/2 and spin-1 particles, although the derivation of the path integral itself looks quite different in those cases. We return to these issues in later chapters of the book.
Next, we need to consider the integration measure. To this end, let us make a change of variables from x(τ ), τ ∈ (0, βℏ), to the Fourier components an, bn. As we have seen, the independent variables are a 0 and {an, bn}, n ≥ 1, whereby the measure becomes
Dx(τ ) =
∣∣det
[
δx(τ ) δxn
]∣∣
∣∣ da 0
[ ∏
n≥ 1
dan dbn
]
The change of bases is purely kinematical and independent of the potential V (x), implying that we can define
C′^ ≡ C
∣∣det
[
δx(τ ) δxn
]∣∣
and regard now C′^ as an unknown coefficient.
Making use of the Gaussian integral
−∞ dx^ exp(−cx (^2) ) = √π/c, c > 0, as well as the above
integration measure, the expression in eq. (1.37) becomes
Z = C′
−∞
da 0
−∞
[ ∏
n≥ 1
dan dbn
]
exp
[
mT ω^2 a^20 − mT
n≥ 1
(ω n^2 + ω^2 )(a^2 n + b^2 n)
]
= C′
2 π mT ω^2
∏^ ∞
n=
π mT (ω n^2 + ω^2 ) , ωn = 2 πT n ℏ
The remaining task is to determine C′. This can be achieved via the following observations:
- Since C′^ is independent of ω (which only appears in V (x)), we can determine it in the limit ω = 0, whereby the system simplifies.
- The integral over the zero mode a 0 in eq. (1.50) is, however, divergent for ω → 0. We may call such a divergence an infrared divergence: the zero mode is the lowest-energy mode.
- We can still take the ω → 0 limit, if we momentarily regulate the integration over the zero mode in some way. Noting from eq. (1.45) that
1 βℏ
∫ (^) βℏ
0
dτ x(τ ) = T a 0 , (1.52)
we see that T a 0 represents the average value of x(τ ) over the τ -interval. We may thus regulate the system by “putting it in a periodic box”, i.e. by restricting the (average) value of x(τ ) to some (large but finite) interval ∆x.
With this setup, we can now proceed to find C′^ via matching.
“Effective theory computation”: In the ω → 0 limit but in the presence of the regulator, eq. (1.50) becomes
ωlim→ 0 Zregulated^ =^ C′
∆x/T
da 0
−∞
[ ∏
n≥ 1
dan dbn
]
exp
[
−mT
n≥ 1
ω^2 n(a^2 n + b^2 n)
]
= C′^ ∆x T
∏^ ∞
n=
π mT ω^2 n , ωn =^2 πT n ℏ
“Full theory computation”: In the presence of the regulator, and in the absence of V (x) (implied by the ω → 0 limit), eq. (1.27) can be computed in a very simple way:
lim ω→ 0 Zregulated =
∆x
dx 〈x|e−^ 2 pmTˆ^2 |x〉
=
∆x
dx
−∞
dp 2 πℏ 〈x|e
− 2 pmTˆ^2 |p〉〈p|x〉
∆x
dx
−∞
dp 2 πℏ e
− 2 pmT^2 〈x|p〉〈p|x〉 ︸ ︷︷ ︸ 1 = ∆x 2 πℏ
2 πmT. (1.54)
Matching the two sides: Equating eqs. (1.53) and (1.54), we find the formal expression
C′^ =
T
2 πℏ
2 πmT
∏^ ∞
n=
mT ω^2 n π
Since the regulator ∆x has dropped out, we may call C′^ an “ultraviolet” matching coefficient.
With C′^ determined, we can now continue with eq. (1.51), obtaining the finite expression
Z =
T
ℏω
∏^ ∞
n=
ω n^2 ω^2 n + ω^2
T
ℏω
n=
[
1 + (ℏω/ n^22 πT^ )^2
] . (1.57)
Making use of the identity sinh πx πx =
∏^ ∞
n=
x^2 n^2
we directly reproduce our earlier result for the partition function, eq. (1.17). Thus, we have managed to correctly evaluate the path integral without ever making recourse to eq. (1.36) or, for that matter, to the discretization that was present in eqs. (1.32) and (1.35).
Let us end with a few remarks:
- In quantum mechanics, the partition function Z as well as all other observables are finite functions of the parameters T , m, and ω, if computed properly. We saw that with path integrals this is not obvious at every intermediate step, but at the end it did work out. In quantum field theory, on the contrary, “ultraviolet” (UV) divergences may remain in the results even if we compute everything correctly. These are then taken care of by renormal- ization. However, as our quantum-mechanical example demonstrated, the “ambiguity” of the functional integration measure (through C′) is not in itself a source of UV divergences.
- It is appropriate to stress that in many physically relevant observables, the coefficient C′ drops out completely, and the above procedure is thereby even simpler. An example of such a quantity is given in eq. (1.60) below.
- Finally, some of the concepts and techniques that were introduced with this simple example — zero modes, infrared divergences, their regularization, matching computations, etc — also play a role in non-trivial quantum field theoretic examples that we encounter later on.
With the relations ˆa†|n〉 =
n + 1|n+ 1〉 and ˆa|n〉 =
n|n− 1 〉 we can identify the non-zero matrix elements, 〈n|ˆaˆa†|n〉 = n + 1 , 〈n|ˆa†aˆ|n〉 = n. (1.67)
Thereby we obtain
G(τ ) =
mω sinh
( (^) βℏω 2
exp
βℏω 2
n=
e−βℏωn
[
e−ωτ^ + n
e−ωτ^ + eωτ^
)]
where the terms are quickly evaluated as geometric sums,
∑^ ∞ n=
e−βℏωn^ =
1 − e−βℏω^
∑^ ∞
n=
ne−βℏωn^ = −
βℏ
d dω
1 − e−βℏω^
e−βℏω (1 − e−βℏω)^2
In total, we then have
G(τ ) = ℏ 2 mω
1 − e−βℏω
)[^ e−ωτ 1 − e−βℏω^
e−ωτ^ + eωτ^
) (^) e−βℏω (1 − e−βℏω)^2
]
2 mω
1 − e−βℏω
[
e−ωτ^ + eω(τ^ −βℏ)
]
2 mω
eωτ^ + eω(βℏ−τ^ ) eβℏω^ − 1
=
2 mω
cosh
[( (^) βℏ 2 −^ τ
ω
]
sinh
[
βℏω 2
] . (1.70)
As far as the path integral treatment goes, we employ the same representation as in eq. (1.50), noting that C′^ drops out in the ratio of eq. (1.61). Recalling the Fourier representation of eq. (1.45),
x(τ ) = T
a 0 +
∑^ ∞
k=
[
(ak + ibk)eiωk^ τ^ + (ak − ibk)e−iωk^ τ
]}
x(0) = T
a 0 +
∑^ ∞
l=
2 al
the observable of our interest becomes
G(τ ) =
x(τ )x(0)
da 0
∫ n≥^1 dan^ dbn^ x(τ^ )^ x(0) exp[−SE^ /ℏ] da 0
n≥ 1 dan^ dbn^ exp[−SE /ℏ]^
At this point, we employ the fact that the exponential is quadratic in a 0 , an, bn ∈ R, which immediately implies
〈a 0 ak〉 = 〈a 0 bk〉 = 〈akbl〉 = 0 , 〈akal〉 = 〈bkbl〉 ∝ δkl , (1.74)
with the expectation values defined in the sense of eq. (1.73). Thereby we obtain
G(τ ) = T 2
a^20 +
∑^ ∞
k=
2 a^2 k
eiωk^ τ^ + e−iωk^ τ^
where
〈a^20 〉 =
da 0 a^20 exp
− 12 mT ω^2 a^20
da 0 exp
− 12 mT ω^2 a^20
mω^2
d dT
[
ln
da 0 exp
mT ω^2 a^20
)]
mω^2
d dT
[
ln
2 π mω^2 T
]
mω^2 T
〈a^2 k〉 =
dak a^2 k exp
[
−mT (ω^2 k + ω^2 )a^2 k
]
dak exp [−mT (ω^2 k + ω^2 )a^2 k]
2 m(ω k^2 + ω^2 )T
Inserting these into eq. (1.75) we get
G(τ ) = T m
ω^2
∑^ ∞
k=
eiωk^ τ^ + e−iωk^ τ ω^2 k + ω^2
= T
m
∑^ ∞
k=−∞
eiωk^ τ ω k^2 + ω^2 , ωk =^2 πT k ℏ
There are various ways to evaluate the sum in eq. (1.78). We encounter a generic method in sec. 2.2, so let us present a different approach here. We start by noting that ( − d
2 dτ 2
G(τ ) = T m
∑^ ∞
k=−∞
eiωk^ τ^ = ℏ m δ(τ mod βℏ) , (1.79)
where we made use of the standard summation formula
k=−∞ eiωk^ τ^ =^ βℏ^ δ(τ^ mod^ βℏ).^2 Next, we solve eq. (1.79) for 0 < τ < βℏ, obtaining ( − d^2 dτ 2
G(τ ) = 0 ⇒ G(τ ) = A eωτ^ + B e−ωτ^ , (1.80)
where A, B are unknown constants. The solution can be further restricted by noting that the definition of G(τ ), eq. (1.78), indicates that G(βℏ − τ ) = G(τ ). Using this condition to obtain B, we then get G(τ ) = A
[
eωτ^ + eω(βℏ−τ^ )
]
The remaining unknown A can be obtained by integrating eq. (1.79) over the source at τ = 0 and making use of the periodicity of G(τ ), G(τ + βℏ) = G(τ ). This finally produces
G′((βℏ)−) − G′(0+) =
m ⇒ 2 ωA
eωβℏ^ − 1
m
which together with eq. (1.81) yields our earlier result, eq. (1.70).
The agreement of the two different computations, eqs. (1.60) and (1.61), once again demonstrates the equivalence of the canonical and path integral approaches to solving thermodynamic quantities in a quantum-mechanical setting.
(^2) “Proof”: ∑∞ k=−∞ eiωk τ (^) = 1 + limǫ→ 0 ∑∞ k=1[(ei^2 βπτℏ −ǫ)k (^) + (e−i^2 βπτℏ −ǫ)k (^) ] = limǫ→ 0
[ 1 1 −ei^2 βπτℏ^ −ǫ^
− 1 1 −ei^2 βπτℏ^ +ǫ
] .
If τ 6 = 0 mod βℏ, then the limit ǫ → 0 can be taken, and the two terms cancel against each other. But if (^2) βπτℏ ≈ 0,
we can expand to leading order in a Taylor series, obtaining limǫ→ 0
[ 2 πτi βℏ +iǫ^ −^ 2 πτi βℏ −iǫ
] = 2πδ( (^2) βπτℏ ) = βℏ δ(τ ).
