Questions tagged [thermal-field-theory]

Thermal Field Theory or Finite Temperature Field Theory deals with of methods to calculate expectation values of physical observables of a Quantum Field Theory at finite temperature.

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Product of delta functions in fermion self-energy at finite temperature

In the calculation of the fermion self-energy at finite temperature, there seems to be a term containing the product of two delta functions which when combined equal zero, however I fail to see why ...
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Decay of metastable state in classical statistical mechanics

Suppose a classical system at temperature $T$ with one variable $m$ and a free energy $F(m)$ having a metastable and a stable minimum. Suppose the system is in the metastable equilibrium at $t=0$. My ...
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Questioning a calculation in Kapusta / Infinite entropy of a Fermi gas?

I am going through Kapusta's calculation of the free energy of a Fermi gas, and I find one of his steps dubious (and if I'm right, it would mean the free energy of a Fermi gas is either infinite or ...
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Coherent States and Temperature for Scalar QFT with Source

This is a follow-up question on a question I previously asked, namely Coherent states and thermal properties. The authors of the article I am referring to in the previous question (Thermodynamics of ...
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Temperature of quantum fields and periodicity

I have read this PSE post Finite Temperature Quantum Field Theory, saying that In a QFT at finite temperature, we consider the Euclidean time to be periodic, i.e. we consider a theory on the manifold ...
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Taylor expansion in momentum integral

On Ashok Das' book "Finite temperature field theory", page 21, the book introduces the thermal mass correction to scalar field. $$ \begin{aligned} \Delta m^2 & =\Delta m_0^2+\Delta m_T^2 ...
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How to derive the Matsubara correlator?

So the Matsubara correlator for either bosons or fermions is given by $$G(i \omega_n) = 1/(i \omega_n - \epsilon_k ), \quad (1)$$ with $\epsilon_k$ being the single particle energy and $\omega_n$ ...
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Troubles with Matsubara sum

In appendix C of Quantum Physics in One Dimension of Thierry Giamarchi, it is claimed that (See (C.22)) after performing the Matsubara sum over the bosonic frequencies $\omega_n=2\pi n/\beta$ in $$\...
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Physical interpretation of thermal 2-point function in QFT

Let $\phi$ be a scalar field, $\rvert \psi_i \rangle$ a set of multiparticle states living in the Fock space of the theory indexed over the naturals, with definite 4-momentum. Let $$\rho_i = \frac{e^{-...
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The Quantum Statistical Average of the Energy-Momentum Tensor

Here: https://arxiv.org/abs/1009.3521 and here: https://arxiv.org/abs/1410.6332 as well as elsewhere, the quantum statistical average of the energy-momentum tensor is taken to be \begin{equation} \...
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How is the path-integral over a spatially finite region calculated?

The partition function for a system in the path-integral formalism is given by \begin{equation} \mathcal{Z}=\int\mathcal{D}\psi\mathcal{D}\psi^{\dagger}{e^{\int_0^{\beta}d\tau\int_Vd^3x\mathcal{...
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Issue with path integrals for the partition function

I was going through Kapusta and Gale "Finite temperature Field theory" In chap 2, Eq. 2.24, they need to do the path integral $$Z = Lim_{N-> \infty} \left (\prod_{i=1}^{N} \int_{-\infty}^{...
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Evaluation of thermal average with path integral

I want to evaluate the thermal average $$<\hat{\phi}(0)\hat{\phi}(0)\hat{\phi}(x)\hat{\phi}(x)>$$ with the path integral. $\phi$ is a real scalar field. In general: $$<\hat{\phi}(0)\hat{\phi}(...
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Relation of Wick theorems

In the context of quantum stat mech it is common to use Wick's theorem to refer to the factorisation $$ \langle f_1 f_2 f_3 \cdots f_N\rangle = \sum_{\text{pairings}\, \pi} (\pm 1)^{|\pi|} \langle f_{\...
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Partition function for bosons with path integral

In this book the partition function for bosons is defined in eq. 2.17 as: $$Z=\mathrm{Tr}[e^{-\beta (H-\mu_i N_i)}]=\sum_a\int d\phi_a\langle\phi_a|e^{-\beta(H-\mu_i N_i)}|\phi_a\rangle$$ The ...
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Harmonic oscillator propagator in Euclidean time

I'm following Nastase's book on Quantum Field Theory but this question is just about quantum mechanics in the path integral formalism. In chapter 8 he considers the propagator equation for a harmonic ...
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Fourier transform of Wick rotated functions

I am learning the imaginary time formalism of thermal field theory / reviewing the Euclidean formalism of quantum field theory. One thing that appears to be left implicit in many treatments is a ...
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Doubts about the "periodicity trick" to compute temperature

The "periodicity trick" is a mysterious way to compute some sort of temperature associated to a Rindler-like spacetime. Suppose there exist coords $R\in(0,\infty), \eta\in(-\infty,\infty)$ ...
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How does spontaneous symmetry breaking (SSB) happen?

I've just finished studying for an exam on the Standard Model (so electroweak theory and symmetry breaking) and I can't figure out how this question never crossed my mind. I'm now studying the QCD ...
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$S$-matrix element for real photon production

In this book (Thermal field theory by Bellac) on page 109 the $S$-matrix element for the transition from an initial state to a final state plus photon $(i)\to(f,\gamma)$ is given: $$S_{fi}^{(\lambda)}(...
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Temperature of AdS-Schwarzchild black holes? How physics is the same for different temperatures?

The five dimensional Schwarzchild-AdS black brane's metric is given by $$ ds^2_5=-\left(\frac{r}{L}\right)^2h(r)dt^2+\frac{dr^2}{\left(\frac{r}{L}\right)^2h(r)}+\left(\frac{r}{L}\right)^2(dx^2+dy^2+dz^...
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Above the electroweak scale, is the Higgs ever effectively massless? (thermal field)

So before the Higgs attains a vev (i.e., above TeV scales), does the Higgs doublet become effectively massless? The Higgs doublet $\Phi$ (not the physical higgs about the nonzero vev) gains an ...
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Quantum field theory at finite temperature

Is there any notion of causality in Quantum field theory at finite temperature?
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Yukawa interaction in QM (0+1D field theory)

This is a question about considering a simple ordinary quantum mechanics system from a quantum field theory perspective. Out of necessity the setup describing the problem is fairly long, but the ...
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E. Fradkin on thermal propagator of free scalar field

In his lecture, E. Fradkin performs a Matsubara sum to show that the finite temperature contribution to the thermal propagator of the free scalar field contains the Bose-Einstein factor (see 5.209 - ...
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Wick rotation on Ward identities

I'm having trouble performing a Wick rotation back to Minkowski spacetime ($\eta_{\mu\nu}=(-1,1,1,\dots)$), following page 19 in the lecture notes here by C.P. Herzog. I have this expression (equation ...
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Matsubara sum with log term [closed]

How do I compute the Matsubara sum $$\sum_n \log\left(-i\omega_n +\frac{k^2}{2m}+\mu\right)?$$ If I have sums like $\sum_n \frac{1}{i\omega_n -m}$, I can sum it up by calculating the sum of residues ...
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Broadening of spectral function: interaction and temperature effect

Consider a non-interacting fermion system with Hamiltonian \begin{equation} H = \sum_{\nu}\epsilon_{\nu}c^{\dagger}_{\nu}c_{\nu}, \end{equation} where $\nu$ is some single-particle quantum number. It ...
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General expression of time-ordered thermal average Green's function does not reproduce non-interacting limit (Fetter ch. 31 Eq. (31.24))

Hi I am going through Fetter's Quantum Theory of Many-Particle Systems Dover Edition. In ch. 31 he computed the relation between $\bar{G}(\mathbf{k},\omega)$, ${\bar{G}}^{R}(\mathbf{k},\omega)$ and $\...
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Green's function in Thermal Field Theory

Background information Let $\beta$ be the inverse temperature 1/T, and $H$ be the Hamiltonian. $H = H_0 + H_I$, where $H_0$ is the free Hamiltonian. Also $S(\beta) = e^{\beta H_0}e^{-\beta H}$ Let $\...
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Partition function for a system in local thermal equilibrium

For a system in equilibrium, the partition function is standard. But if the system is in local thermal equilibrium but stationary (i.e. zero or negligible time variation), but the temperature varies ...
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Derivation of Thermally averaged cross sections

In many sources discussing neutrino decoupling I find the following claim: "The thermally averaged rate of weak interactions is given by: $\Gamma = n \langle\sigma |v|\rangle$, where $\langle\...
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What does the Temperature of a QFT physically mean?

In elementary statistical mechanics, one can think of temperature as arising from the average kinetic energy of particles in the ensemble. Is there a similar way to think about the temperature of a ...
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Why is Euclidean Time Periodic?

I've been reading a bit about finite temperature quantum field theory, and I keep coming across the claim that when one Euclideanizes time $$it\to\tau,$$ the time dimension becomes periodic, with ...
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Which is the state of the art of relativistic finite-density QFT?

1 - QFT is typically used to calculate relativistic scattering. The ground state of relativistic QFT is the vacuum, and it is invariant state under Lorentz transformations. This is not the case when ...
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How does a thermal propagator work?

I am looking at a propagator in the Hubbard model (in the strong coupling limit) and my timescale is $\beta$. I see that for longer (imaginary) times $\tau$, the particle can propagate further away. ...
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How to calculate expectation value of exponentiation of number operator for coherent state?

I consider a quantum harmonic oscillator and regard $a$ and $a^\dagger$ is ladder operators. Let $|0\rangle$ be a vacuum, and a coherent state $|\alpha\rangle$ is defined as the eigenstate of the ...
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Temperature reduction in 4D QED

I would like to find references for the following topic. Consider QED with non-zero temperatures, which is naively constructed by Wick rotation. Then, consider the case of high temperatures, $\beta\...
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How does QED at a finite temperature differ from QED at zero temperature?

Currently, I do not have any knowledge of finite temperature field theory. But I have learnt ordinary QFT calculations and I am reasonably familiar with Statistical mechanics. With this background, I ...
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Conceptual Problems in Understanding Thermal Field States in nonrelativistic QED

I have conceptual problems understanding the notion of thermal states in the context of nonrelatistic (cavity) QED. My main problem is the definition of temperature associated to the (quantized) ...
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Cross-sections at zero temperature and high temperature for a process and its reverse

If the Feynman amplitude for a $2-2$ forward scattering $ab\to cd$ is denoted by $\mathcal{M}_{ab\to cd}$ and that of the reverse scattering process, $cd\to ab$, is denoted by $\mathcal{M}_{cd\to ab}$...
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Why is chaos a common property of thermal systems?

https://arxiv.org/abs/1811.06949 pg 3 mentions that chaos is a common property of thermal systems. Can someone please explain why that is? While looking at [1], I found that indeed most examples ...
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Free parameter in Bose Einstein Condensate

In Kapusta and Gale's Finite-Temperature Field Theory book, BEC is derived for a complex scalar by Fourier expanding $$\phi _1 = \sqrt2 \zeta \cos \theta + \sqrt{\frac{\beta}{V}}\sum_{n,\bar p}e^{i(\...
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Retarded Green's function is the same as time-ordered Green's function at zero temperature for $\omega>0$

Consider the equilibrium time ordered and retarded Green's function for two operators $A,B$, defined as follows $$ G^T(t,t') \equiv -i\langle T A(t)B(t') \rangle=-i\theta(t-t')\langle A(t)B(t')\rangle ...
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Confusion about trace in the vertex term of Lagrangian

I was reading through Mariano Quirós's lecture notes titled "Finite Temperature Field Theory and Phase Transitions". In Sec. 1.2, the author is calculating the one-loop effective potential at $T=0$. ...
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Understanding Matsubara summation

I'm trying to understand matsubara summation. Let us say I have $f(i\omega) = 1$. Obviously, the matsubara summation $\sum_{\omega_n} f(i\omega_n)$ diverges. So, I use a weighing function. Let us ...
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How does one modify the decay width of a particle (QFT/Thermal Field theory style) when a particle is travelling through matter

I believe a particle's decay width/rate should depend on whether they are in matter or vacuum, but am unsure of where to find a prescription describing this phenomena. Please could someone point me in ...
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Guessing the temperature dependence of a decay rate $\Gamma(A\to B+B)$

For a two-body decay of the form $$A\to B+B$$ if the interaction strength controlling the decay is $\lambda$, the Feynman amplitude $\mathcal{M}$ will contain a factor of $\lambda$ from the vertex ...
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Classical chaos at finite temperature

Is there any finite temperature generalization of classical chaos? In quantum chaos, at least with regards to out-of-time-order correlators, the generalization is clear - one simply takes a thermal ...
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Reference for Feynman diagram technique(position space) in Thermal Field Theory

I am trying to study perturbative expansion of Sachdev-Ye-Kitaev model, where I know that the dominant terms are the Melonic diagrams. I am interested in seeing how perturbative corrections affect the ...
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