Questions tagged [quantum-field-theory]

Use this tag for questions about quantum field theory in theoretical/mathematical physics. Quantum Field Theory is the theoretical framework describing the quantization of classical fields allowing a Lorentz-invariant formulation of quantum mechanics. Associate with [tag:mathematical-physics] if necessary.

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0answers
52 views

In what situations are Path Integrals well-defined?

In physics I have come across contexts where apparently path integrals are well-defined, and others where they are not. However I have no clear understanding of when and why they succeed or fail to be ...
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45 views

Introduction to representation theory? [closed]

I’m studying quantum field theory and have reached the point where I need to understand representation theory for a deeper understanding (specifically for why certain equations give spin-0, 1/2, and 1 ...
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Transformation of line element in Minkowski space under infinitesimal coordinate variation

I'm trying to understand the following problem: We are looking at an infinitesimal coordinate transformation $$ x^\mu \rightarrow x^\mu + \epsilon u^\mu(x), \space \epsilon \rightarrow 0 $$ and we are ...
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58 views

Understanding gauge transformations and relation to Lie Groups?

I am starting to study gauge theory. I have a background in basic group theory, multivariable calculus, and the idea of symmetry in relation to group theory. I am trying to understand why the ...
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26 views

Laplace transformation in quantum field theory

I am doing some quantum field theory (physics) calculations with the fermion mass term, and it is making the calculation much more challenging. Say I have $$F(t,m) = \int_0^1 dx \frac{1}{1-x}\frac{1}{(...
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60 views

Derivative after applying residues theorem, that removes original poles

I have a question related to a typical integration in particle physics. Suppose one has a function $f(t) = \dfrac{i}{t^2-\omega^2 }$ with $t, \omega \in \mathbb{R}$ and $\omega>0$. I am interested ...
2
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1answer
65 views

What mathematical skills are needed to learn Lagrangian Field Theory (building up to QFT)?

I want to start teaching myself Lagrangian Field Theory. I can do multivariable calc, tensor calc, Lagrangian mechanics, and some calculus of variations. Are there other math fields I should study ...
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102 views

interpretation of combinatorial term of partition function

I'm currently reading A. Zee "Quantum Field Theory in Nutshell". There is a problem called baby problem which is to evaluate following partition function. $$ Z(\lambda ,J) := \int_{-\infty}^{...
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16 views

Gradient of Free Scalar Field which is an Integral

I have to calculate: $$\frac{1}{2}\int d^3x (\nabla\phi)^2.$$ Where $$\phi = \int d^3k \frac{1}{(2 \pi)^3 \sqrt{2\omega_k}} \left(a(\vec{k})e^{-i(\omega_kx_o-\vec{k}\cdot \vec{x})}+a^{\dagger}(\vec{k})...
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Invariance of Lagrangian density under gauge transformation

In a problem in my QFT course i am asked to write down the most general posible Lorentz invarient Lagrangian density for a vector field (at most quadratic in the field and in the derivatives) and to ...
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36 views

Are affine time-distortions the only ones that preserve the positive-frequency property?

Suppose that a function $f(t)$ can be written $$ \newcommand{\reals}{\mathbb{R}} f(t)=\int_0^\infty d\omega\ F(\omega)e^{-i\omega t} $$ for some absolutely integrable function $F(\omega)$. This is ...
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1answer
42 views

Functional derivative of four-gradient

I am begining a course in QFT, and am starting with the topic of functional derviatives. In the problem set given, we are asked to calculate the functional derivative of $$F[\phi] = \partial_{x}\phi(x)...
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50 views

Name of the proceedure used to obtain the Dirac equation?

What follows below is a brief description of the procedure Dirac used to obtain the Dirac equation. For my question, which is about this procedure, see the last paragraph. Historically, Dirac was ...
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12 views

Complete characterization of local monomials

I'm reading Zavialov's book on QFT and there's a statement there, that I was interested in finding out how to prove. The statement is as follows: If $[A(x), j_{\{\lambda\}}(y)]=0$ for $(x-y)^2<0$ ...
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53 views

QFT: Interacting Lagrangian and associated Momentum Space Formulas

Consider a real scalar field with the free part of the Lagrangian $L_0 = −\frac12(\partial \varphi)^2 - \frac12(m^2 \varphi^2),$ and an interaction term $L_1 = \frac12g(φ^2)$. What are the momentum-...
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57 views

Formula for calculating symmetry factors for Feynman diagrams?

In a homework exercise for a QFT course I'm taking, we have been tasked with calculating the integral $$ Z(g)=\int_{-\infty}^{+\infty} dx\, e^{-\frac{1}{2}x^2 + \frac{g}{3!}x^3} $$ up to order $g^4$ ...
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16 views

Following a calculation of entropy in the first quantization scheme

I'm trying to follow the computations of example 5.1 in this paper. To begin with they have a symplectic Hilber space $(\mathcal{K},\tau,\sigma)$, where $(\mathcal{K},\tau)$ is a separable Hilbert ...
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38 views

Motivation behind the conjugate Schrödinger equation

What is our motivation behind the conjugate Schrödinger equation (SE), mathematically a conjugate complex scalar is a "projection" of that complex scalar on its opposite quadrant. But what's ...
2
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1answer
70 views

Normalization of One-Particle States using Dirac Delta Function

I am attempting to understand the normalization of one-particle state $|\textbf{p}\rangle \propto a_{\textbf{p}}^\dagger$ in the context of Klein-Gordon field quantization from Peskin & Schroder's ...
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1answer
53 views

How to Solve this Complex Integral

The following integral $$\frac{2\pi}{ir}\int_{0}^{\infty}ke^{-i\omega_kt}(e^{ikr}-e^{-ikr})dk$$ arises in finding the probability amplitude: $\langle x|e^{-i\hat{H}t}|x=0\rangle$. Here $\vec{x} = r\...
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1answer
36 views

Lorentz Invariance of the Choice of $p_0 = \pm E_{p}$

I'm following David Tong's notes on Quantum Field Theory and I'm struggling with proof of the Lorentz invariance of $\int d^3 p/(2E_p)$ given on page 32. The proof contains the line "Solving for ...
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1answer
94 views

Is the Fourier Transform of a quantum field well-defined?

Let there be a separable Hilbert space $\mathcal{H}$ and let $\hat\phi(t,\vec{x})$ be a bounded linear operator on $\mathcal{H}$ for every $t\in\mathbb{R},\vec{x}\in\mathbb{R}^3$. There exists a ...
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23 views

Position of the poles of two point function .

Let $\phi$ be a quantum scalar field and suppose we have $\phi=\psi+F(\psi)$. The relation between the two point function of $\phi$ and $\psi$ is given by $$ \begin{aligned} \langle 0|T(\phi(x) \phi(...
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1answer
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Question on complexification of $\mathfrak sl(2,\Bbb C)$

As six generators of the real Lie algebra $\mathfrak sl(2,\Bbb C)_\Bbb R$ I can use the Pauli matrixes as follow: $X_1=\frac{1}{2} \sigma_1, X_2=\frac{1}{2} \sigma_2, X_3=\frac{1}{2} \sigma_3$ $Y_1=\...
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Question about Lorentz group related Lie algebra [duplicate]

I am self-studying Lie algebra for the Lorentz group, and I would like to double-check if my understandings are correct, cause on many physics books the most of the time is not clearly stated if he is ...
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Integrals of functions with poles on the real axis (Sokhotski–Plemelj theorem) in the sense of Distributions/Generalized Functions.

The Sokhotski–Plemelj theorem [Wikipedia link] states that $$ \lim_{\epsilon \to 0 } \int_{\mathbb R} dz \; \frac{f(z)}{z \pm i \epsilon} = \mp i \pi f(0) + \mathcal{P} \int dz \; \frac{f(z)}{z} \;, ...
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1answer
138 views

Integrating product of Dirac deltas and step functions

I have the following integral $$\int d^4\boldsymbol{x}' \,\delta\big[(\boldsymbol{x}-\boldsymbol{x'})^2+\alpha^2\big]\,\Theta(-x_0+x'_0)\,\delta\big[(\boldsymbol{x'})^2+\alpha^2\big]\,\Theta(-x_0'),\...
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1answer
17 views

How to derive a Lorentz covariant, generic solution for the system in a Klein-Gordon field.

Below, I am reading through how to quantise a complex scalar field, beginning with an example of the Klein-Gordon field. After arriving to the generic solution to the system corresponding to equation ...
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29 views

Understanding the operators required to derive a more generalised form of the Schrodinger equation.

The following is an excerpt from my lecture notes, aiming to derive a more general form of the Schrodinger equation. There has been little reason for the introduction of the commutation relations ...
4
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1answer
85 views

Addition of Permutations

Sorry, if the title doesn't provide any clarity, but I didn't really know how to call it. Anyways, I've been studying quantum field theory from Blundell's book and during the derivation of the formula ...
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45 views

What does this definition of a Weyl star algebra in Spectral theory and QM by Moretti, 2013 mean?

Here's the definition I find very difficult to understand, in particular the last part in boldface. I don't understand what is means to take "combinations of" W(u) (is it linear combinations?...
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60 views

Free massive scalar field partition function in QFT?

I have also asked this question here. I am posting it on this forum as well in order to increase the number of people who see this question. Consider the (euclidean) path integral for the free massive ...
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1answer
258 views

Show $\lim\limits_{t\to\infty}\Bigg|\sum\limits_{n=0}^\infty\Theta(t-nR)\frac{(\Gamma(t-nR))^n}{n!}e^{-\Gamma(t-nR)}\Bigg|^2=\frac1{(1+\Gamma R)^2}$

I have encountered the following problem while studying non-Markovian effects in real-time dynamics of open quantum systems. In particular, I was studying a system comprised of two qubits (qubit is a ...
2
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1answer
45 views

A condition for the vacuum generating functional

In Theorem 1 of this paper Segal stablish a relation between states and generating functionals. He assert that in order to μ be a generating functional must satisfy Then, as an example he show the ...
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0answers
15 views

Request for expository articles on supersymmetric geometry.

Various kinds of supersymmetric QFTs are studied in the physics literature. A typical physics talk describes a Lie "superalgebra" by a huge list of operators (with many supercharges, ...
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37 views

Equivalence between the equations of motion derived from the two variants of Nambu-Gotto action...

I am trying to show two equations of motion derived from 2 variants of Nambu-Goto action are equivalent... We have the Nambu-Goto action in terms of the induced matrix as $$S=-T\int d^2 \sigma\sqrt{-\...
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0answers
201 views

Reading list recommendation for a hep-ph student to start studying QFT at a more mathematically rigorous level?

Edition On July 15 2021, the description of the question has been considerably modified to meet the requirement of making this question more OP-independent and thus more useful for general readers. ...
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59 views

Bestiary of well defined QFTs

In trying to understand something about Quantum Field Theories and especially the mathematics behind them (or lack thereof) I have a hard time harvesting a handful of well defined examples (other than ...
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44 views

How to solve this integral for three-point correlation function?

I need to solve the following integral as part of the computation of a three-point correlation function in a charged massless quantum field theory, where I can also assume conformal invariance. $$ \...
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1answer
39 views

How do I find the combined Hilbert space for this Hamiltonian?

I knew that the operators in the folwing Hamiltonian act in different Hilbert spaces, so I cannot just multiply them. $$\eqalign{ & H = g\left[ {\left( {a\sigma _1^ + + {a^\dagger }\sigma _1^ - ...
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Motivation for quantum cohomology rings

I can't seem to find a good source for the motivation for defining the big quantum cohomology ring with its quantum product. Collecting the Gromov-Witten invariants in a generating function seems like ...
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0answers
48 views

What is the cardinality of the set of possible wavefunctions $\Psi: (\mathbb{R}^3 \to \mathbb{R}) \to \mathbb{C}$?

In QFT (quantum field theory), a wave-function $\Psi$ takes in a scalar field ($\phi: \mathbb{R}^3 \to \mathbb{R}$) as input, and returns a complex number ($\mathbb{C}$) as output. Thus: $$\Psi: (\...
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1answer
63 views

Feynman propagator as a distribution

There is something probably simple which is confusing me. In his Quantum Field Theory, Folland defines (or derives) the Feynman propagator as $$\Delta_F(t, \mathbf{x}) = i \int_{\mathbb{R}^3} \frac{e^{...
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1answer
26 views

Definition of anticommuting Mean Field Theory and real Grassmann fields

These words come from this article https://arxiv.org/abs/2008.04361, page 6. My question is what is anticommuting Mean Field Theory which appears in the equation (1.1), and what is real Grassmann ...
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37 views

S. Weinberg, “The Quantum theory of fields: Foundations” (1995), Eq. (9.2.15)

In Weinberg's book The Quantum Theory of Fields, volume 1 on p.387, there is a Fourier transform as: $$\mathscr{E}(\mathbf{x},\mathbf{y})=(2\pi)^{-3}\int{\mathrm{d}^3p\,e^{i\mathbf{p}\cdot (\mathbf{x}-...
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1answer
92 views

Comparing Hamiltonians - Quantum harmonic oscillator

For standard 1D quantum harmonic oscillator we have $H\psi = E_n\psi$ with $E=(n+\frac{1}{2})\hbar\omega$ and $H = \frac{P^2}{2m} + \frac{1}{2}m\omega^2 X^2$ where $X$ is position operator and $P$ is ...
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0answers
56 views

Variational calculus on Lie algebra valued one forms - Chern-simons theory

The action in the Chern-Simons theory is given as: $$S=\frac{k}{4\pi}\int_M \text{Tr}(A \: \wedge \:dA + \frac{2}{3}A \: \wedge \:A \: \wedge \: A). $$ The wikepida page gives the euler lagrange ...
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2answers
142 views

Matsubara sum arising from QFT and contour integral

In the lecture of E. Fradkin on quantum field theory, an example of Matsubara sum is performed using contour integration (see eq. 5.214 in the lecture). It reads $$ \sum_{n=-\infty}^{\infty} \frac{e^{...
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1answer
53 views

Square of the squeezing operator

What is the square of the squeezing operator $S(z)=\exp[\frac12\left((z(a^{\dagger})^{ 2}−z^\ast a^2\right)]$? I mean, with $z \in \mathbb{R}$, what is $S(Z)S(Z)$? Is there any formula ?
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1answer
57 views

Can't understand how the authors got this equation?

Let us consider a two-qubit system (A and R) initially entangled as given by: $$ \left| {{\Psi _{AR}}} \right\rangle = \alpha \left| {{0_A}} \right\rangle \otimes \left| {{1_R}} \right\rangle + \...

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