2.1. Quantum Field Theory
2.1.4. Thermal field theory & statistical field theory
The Wick rotation not only allows us to define the functional integral more rigorously, but also defines the partition function of classical statistical field theory11
Z“ ż
Dϕe´1{¯hSrϕs „ ż
Dϕe´βHrϕs, (2.47)
where the action describes the static energy functional H of a statistical configurationϕ ind dimensions. Note that the Euclidean time is yet another direction/dimension of the configuration space and must not be confused with temporal evolution. Thus, we may conclude that Euclidean quantum field theory inddimensional spacetime can be identified with classical statistical field theory with Boltzmann weight
e´1{¯hSrϕs „ e´βHrϕs “ e´Hrϕs{kBT ÝÑ ¯h„kBT “β´1, (2.48)
10Compare to the statement of strong quantum fluctuations in the classical limit below (2.22). In fact the semi-classical expansion of the saddle-point method follows the same logic as the perturbative expansion of the interacting field theory.
11In this section we explicitly write out the factors of¯handkBto elucidate the nature of the mapping.
where the temperatureT is identified with the inverse of the Planckian action¯h´1. In general the temperature of the classical statistical field theory isnotrelated to the temperature of the quantum system, but to the dimensionless coupling constant of the quantum theory. Note that the removal of the temporal evolution yields the identification of an action functional with an energy functional
Srϕs “ ż
ddxL “ ż
dτdDx
”
pBτϕpτ,xqq2` p∇ϕpτ,xqq2`Vpϕqı ,
Hrϕs “ ż
ddxE “ ż
dD`1x”
p∇ϕpxqq2`Vpϕqı ,
(2.49)
where the spacetime dimensionality of the quantum field theoryis given byd “ D`1 andD denotes the number of spatial dimensions.
The mapping between Euclidean quantum field theory and classical statistical field theory can be extended to partition functions of quantum statistical systems. This follows directly from the definition of the quantum partition function
Z
“tr e´βH “ÿn
A
nˇˇˇe´βH ˇˇˇnE
, (2.50)
We can retrace steps
ii
andiv
on page 15 of constructing the path integral, but this time with imaginary timeτ running from0toβ and under periodic boundary conditions identifying|qiy ” |nyand|qfy ” |nyby virtue of the identification A
qfˇˇˇe´i{h¯Hptf´tiqˇˇˇqiE
“A
qpτfqˇˇˇe´1{h¯Hpτf´τiqˇˇˇqpτiqE
„A
nˇˇˇ e´βH ˇˇˇnE
, (2.51)
implying
¯
hβ“τf´τi and |qfy “ |qpτfq y “ |qpτf `¯hβq y “ |qiy. (2.52) The periodic boundary conditions (2.52) yield a non-trivial topology12so the quantum partition function is equivalent to an imaginary time evolution over a circleS1with circumference
LT “ ¯h
kBT “¯hβ. (2.53)
Extending the path integral representation to a functional integral representation for a higher dimensional systems, introducing coherent states, we can identify a Euclidean quantum field theory onS1ˆRDwith a quantum statistical system inRD13
Z
“tr e´βH “ż
Dϕexp
„
´
¿
S1
dτ ż
dDxL
. (2.54)
Remarkably, thermal quantum field theory in equilibrium is obtained by compactifying the imag-inary time direction on a circleS1 with radiusLT{2π. As an additional side effect the Fourier
12For fermions one obtains antiperiodic boundary conditions due to the anticommuting nature of fermionic coherent states.
13Note that the coherent states do introduce the properpq9term such that the Hamiltonian density is transformed into the Lagrange density seee.g. [51]
transform of the imaginary time becomes discrete, the so-called Matsubara frequency represen-tation
ϕpτ,xq “ÿ
n
ϕnpxqe´iωnτ, ϕnpxq “ 1 β
żβ 0
dτ ϕpτ,xqeiωnτ. (2.55) where the Matsubara frequencies are given by14
ωn “2πnT, nPZ. (2.56)
The discreteness of the Matsubara frequencies can pose a problem to obtain the proper analytic continuation of the imaginary time correlation functions (e.g. the analytic continuation of a func-tion only defined on discrete points is not unique). Furthermore, for more complex applicafunc-tions the analytic continuation becomes highly non-trivial and might not be feasible at all. Even worse, any approximation made to obtain an analytic continuation might be totally uncontrolled and could in principle lead to spurious real time results. These issues become important when deal-ing with linear response and transport coefficients where we need to look at fluctuations about the equilibrium state, yet the measurements are done in real time (see Section2.2.2).
From the quantum partition function (2.54) we have access to all thermodynamic properties by introducing thermal (equilibrium) averages and employing the statistical density operator describing the quantum statistical canonical ensemble or a quantum system in a mixed state
xϕy “trpρϕˆ q where ρˆ“ 1
Z
e´βH. (2.57)
Note that we can easily include the chemical potential µ by simply shifting the Hamiltonian
H
ÝÑH
´µN
whereN
denotes the number operator (of a species). Thus, we allow the number of certain particle species to fluctuate where the average number is fixed by the respec-tive chemical potential. This is the quantum grand canonical ensemble or a quantum system described by Gibbs states. In Table 2.1 the correspondence between Euclidean quantum field theory and quantum statistical field theory is shown including all “identifications” of the con-cepts we defined in Section2.1.2. Futhermore, we can look at two interesting limits of thermal field theories:• The zero temperature limitT Ñ 0, β Ñ 8effectively removes the time circle S1 since the circumference becomes infinite. Therefore, we recover the quantum field theory over infinite spacetimeRd
• In the high temperature limitT Ñ 8, β Ñ0 the circleS1 shrinks to zero and so we ef-fectively remove the imaginary time dimension completely. The Euclidean quantum field theory is now defined overRD. This is in agreement with the statement that for high temperatures the quantum nature of a system is lost and hence we end up with a classi-cal statisticlassi-cal field theory. In this case we recover the correspondence between Euclidean quantum field theory and classical statistical field theory of the same total dimension.
The last point can be understand physically as the suppression of quantum fluctuations at high temperatures. Following the saddle point method (2.19) the classical action is time independent and the quantum fluctuations up to quadratic order contribute a term quadratic in Matsubara frequencies
Sfluct“ÿ
ωn
ω2nδqpωn,xqδqp´ωn,´xq. (2.58)
14Fermionic Matsubara frequencies readωn“ p2n`1qπTwithnPZ.
Euclidean quantum field theory Quantum statistical systems ddimensional Euclidean spacetime ô Ddimensional spacexPRD
pτ,xq PS1ˆRd´1
Correlation functions ô (generalized) moments
Generating functionallnZ ô Thermodynamic potentials
Ω“ ´β´1ln
Z
Connected correlation functions ô Cumulants
Wick’s theorem ô “Corresponding theorem”
(for moments in Gaussian theory) Perturbative expansion in interactions ô Cumulant expansion of interaction term Table 2.1. Euclidean quantum field theory can be related to statistical field theory for quantum statistical systems. All concepts derived in quantum field theory can be carried over immediately to concepts in statistical physics. In particular the machinery to generate connected correlation functions determines the thermodynamic potentials.
From a mathematical point of view, the formulation of quantum field theory by means of stochastic functional integration puts it on a firm mathematical basis.
If the quantum fluctuation energyωn„Texceeds the characteristic energy scale set by the clas-sical action all non-vanishing Matsubara modes can be neglected. In imaginary time representa-tion the increase of the quantum fluctuarepresenta-tion energy corresponds to the shrinking of the imaginary time intervalτ P r0, βswhich increases the contributions of the gradient terms (through steeper slopes) by Bτδqpτ,xq „ β´1 „ T. This argument is crucial in understanding the interplay of quantum and thermal fluctuations of quantum phase transitions in particular the shape of the quantum critical region,c.f. Figure1.1. As a final remark we may combine the correspondence between classical statistical field theory and Euclidean quantum field theory with the correspon-dence to quantum statistical theories. Thus, we end up with an mapping fromD dimensional quantum systems ontoD`1dimensional classical systems. Moreover, theDdimensional imag-inary time correlation functions are mapped to correlation function in theD`1 dimensional classical field theory. This mapping becomes precise for large correlation lengths such that the discrete microscopic details of the theories are averaged out by a proper renormalization to ob-tain a true effective field theory. In any case the universal thermodynamic properties must not depend on the underlying microscopic model.