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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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References discussing renormalizability in quantum field theory via Sobolev norms

Hawking & Hertog's paper Living With Ghosts has a nice introduction in which the authors discuss the issue of renormalizability of a field theory in terms of Sobolev norms. More specifically, they ...
Níckolas Alves's user avatar
1 vote
2 answers
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Given Green's function, can I find the corresponding operator?

Green's function is the solution to the equation $L G(x;x') = \delta(x-x')$, where $L$ is a linear differential operator. Usually, we want to find the Green's function of a given $L$. Instead, if we ...
Sean's user avatar
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Some details concerning projective representations in Wienberg's book

I have a question from the book "the quantum theory of fields" by S. Weinberg in page 89: How can we get $[U(\Lambda )U(\bar{\Lambda})U^{-1}(\Lambda \bar{\Lambda})]^2=1$ from the fact that ...
Mahtab's user avatar
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Are creation and annihilation operators special?

In Weinberg's The Quantum Theory of Fields,volume I, the author quotes a theorem that left me a bit mystified. He states Any operator $O: \mathscr{H} \rightarrow \mathscr{H}$ may be written $$O=\sum_{...
Lourenco Entrudo's user avatar
2 votes
1 answer
94 views

How should we think about functional differentiation

I've been introduced to the following notion of functional differentiation: Let $F: \Phi \rightarrow \mathbb{R}$ be a linear functional on $\Phi$ (understood as a space of sections over a manifold $M$...
Lourenco Entrudo's user avatar
3 votes
1 answer
122 views

How to calculate $\int_{0}^{1}\operatorname{ln}(ax^2+bx +c) dx$?

In $\ $Matthew D. Schwartzs Quantum Field Theory$\ $ book, p. $302$, he derives next formula $$ {\rm i}\,\mathcal{M}_{\rm loop}(p) = -\,{\rm i}\,\frac{g^{2}}{32 \pi^{2}} \int_{0}^{1}{\rm d}x\ \ln\left(...
Plantation's user avatar
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Fourier transform with and without convolution theorem not equivalent

This is a problem involving Fourier transforming an integral relevant to the computation of Feynman diagrams, which is of the form: $S(r_1,r_2)=\int d^3 r_3 \space v(r_1,r_3)f(r_3,r_2),$ where $v(r_1,...
user2188518's user avatar
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Integration measure for a strange substitution

I have a 2D integral over a momentum vector, i.e. $\int dp_x dp_y$ and the substitution for this is given by $$ \xi = |\vec{p}| + |\vec{p} + \vec{q}| , \, \, \, \eta = |\vec{p}| - |\vec{p} + \vec{q}|$$...
Johnny_T's user avatar
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Converting an integral to hypergeometric function [closed]

I have encountered an integral as follows $$\int_{0}^{1}{\left(k^{2}x^{2}-k^{2}x+m_{2}^{2}+m_{1}^{2}x-m_{2}^{2}x \right)^{\frac{d-4}{2}}}dx$$ Any suggestion how to convert it into a hypergeometric ...
NovoGrav's user avatar
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How to lower bound the quantum conditional entropy?

I am trying to lower bound the quantum conditional entropy $H(X|Y)$ when $X$ and $Y$ are two quantum systems. Classically, it can be done as follows: $$ H(X|Y) = \sum_{y}P_Y(y) H(X|Y=y) \geq \sum_{Y \...
Jaswanthi Mandalapu ee19d700's user avatar
2 votes
0 answers
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What is a transformation (physical) in precise terms?

I’m reading some quantum field theory from physics (David Tong’s lecture note), but I don’t quite understand what it means to perform some transformation $x \rightarrow x’$. I’m trying to view it in ...
Frozer Clark's user avatar
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1 answer
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I am confused about a certain property of spinor transformations that seems to be inconsistent based on my current understanding.

I am considering the transformation of a two dimensional Weyl spinor $\lambda^\alpha$ given by a matrix transformation of the form $p(\lambda)^\beta = \lambda^\alpha m_\alpha^{\ \ \beta}$. Let's say ...
Teddy Baker's user avatar
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Deriving Components of a Quaternion-Based Wave Function

I am currently exploring an intriguing topic related to quaternion-based wave functions and have encountered a mathematical challenge that I hope to get some insights on. The concept is detailed in an ...
p yz's user avatar
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How to evaluate Lorentz invariant integral with complex parameters

For context: there are Schwartz distributions called Pauli-Jordan (or Schwinger) functions which come up in quantum field theory. Given $m>0$, $x_0\in\mathbb{R}$ and $\vec{x}\in \mathbb{R}^3$, $D_m^...
TheEmptyFunction's user avatar
4 votes
1 answer
208 views

Problems differentiating four-vectors to find the equation of motion using Euler-Lagrange equations.

This post directly follows this question and is very similar in nature: Consider the following Lagrangian: $$\mathcal{L}=\frac14\left(\partial_\mu A_\nu-\partial_\nu A_\mu\right)\left(\partial^\mu A^\...
Sirius Black's user avatar
2 votes
1 answer
115 views

Using Euler-Lagrange equations show that the following EoM can be written as $D_\mu D^\mu\phi+m^2\phi + \lambda\left(\phi^\ast\phi\right)\phi=0$

This post is a follow-up to this previous question. Consider the following Lagrangian: $$\mathcal{L}=\frac14\left(\partial_\mu A_\nu-\partial_\nu A_\mu\right)\left(\partial^\mu A^\nu-\partial^\nu A^\...
Sirius Black's user avatar
1 vote
0 answers
46 views

Bijection from (proper) Lorentz group to PSL(2,C)

It is well known that $SL_2(\mathbb{C})$ is the universal cover of $SO^+(1,3)$, see for example the Wikipedia page on the Lorentz group 1. The map goes like this (some of this is not standard ...
TheEmptyFunction's user avatar
3 votes
1 answer
221 views

Show that $\partial_\mu\phi^\ast A^\mu\phi- A_\mu\phi^\ast\partial^\mu\phi=A^\mu\phi\partial_\mu\phi^\ast - A^\mu\phi^\ast\partial_\mu\phi$

The following is loosely related to this question: [...], the most general renormalisable Lagrangian that is invariant under both Lorentz transformations and gauge transformations is $$\mathcal{L}=-\...
Sirius Black's user avatar
5 votes
1 answer
157 views

What is the correct sign for the four-vector potential gauge transform; $A_\mu\to A_\mu\pm\partial_\mu\lambda$ and where does this gauge originate? [closed]

I have three questions regarding the following extract(s), I have marked red the parts for which I do not understand for later reference. The convention followed for the Minkowski metric in these ...
Sirius Black's user avatar
4 votes
1 answer
183 views

Square of the Dirac Delta Shenanigans

The following is a purely mathematical problem, but it often arises when dealing with transition probabilities in Quantum Field Theory. In QED the transition probability amplitudes often contain a ...
Noumeno's user avatar
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Can any meaning be given to this integral?

As an example, consider the Klein-Gordon equation: $$(\partial_t^2 - \Delta + m^2)\psi = 0. \tag{1}$$ In physics one usually starts with an ansantz of plane wave solutions so begin by writing $\psi$ ...
CBBAM's user avatar
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2 votes
1 answer
138 views

Notation for the d'Alembert operator

From the Wikipedia page on the d'Alembert operator it is stated that equivalent ways of writing the d'Alembert operator are as follows, $$\begin{align}\Box =\eta^{\mu\nu} \partial_\nu\partial_\mu = \...
Sirius Black's user avatar
3 votes
0 answers
69 views

Does the order of operators reverse when taking the Hermitian adjoint of a product of many operators?

${}^\zeta$ Defining the vacuum state $\lvert 0 \rangle$ as that where $\hat a(\vec p)\lvert 0 \rangle=0$ and normalizing it as $\langle 0\lvert 0 \rangle = 1$, then we again build the Hilbert space as ...
FutureCop's user avatar
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0 answers
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Solution of the Yang-Baxter equation not coming from quasi-triangular structure

Let $A$ be an associative, unital algebra over a field $\Bbbk$, and let $R \in A \otimes A$ be an invertible element which is a solution of the Yang-Bater equation in $A \otimes A \otimes A$ $$R_{12}...
Minkowski's user avatar
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1 vote
0 answers
60 views

Fourier transformation with a square root-log term

(Note: I posted the exact same question in the physics StackExchange, but to get a breadth of people looking at the problem, I am coming to the Math StackExchange also since, well, it is just an ...
MathZilla's user avatar
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Non-reversible time-dependent wave function - which type of PDE?

Is anyone aware of a time-dependent wave-function that is non-reversible? With non reversible, I mean that the initial state of some wavefunction as solution to some PDE, $\psi_0(x_1,\dots,x_n,t)$ in ...
Superunknown's user avatar
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-1 votes
1 answer
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For eigenvector $Hv= \lambda v$, $(H-\lambda)(\{v\}^{\perp}) \subseteq \{ v\}^{\perp}$ and $H-\lambda$ is invertible on $\{ v\}^{\perp}$?

Literally, let $H : \mathcal{H} \to \mathcal{H}$ be an operator of Hilbert space, or finite dimensional inner product space. Assume that $H$ has an eigenvector $\lambda$ with unit eigenvector $v$. ...
Plantation's user avatar
  • 2,698
2 votes
0 answers
77 views

Proof explanation for $\left[\hat a(\vec p)^\dagger\hat a(\vec p), \hat a(\vec q)^\dagger\hat a(\vec q)\right]=0$

We have the usual commutation relations for creation/annihilation operators in QFT, $$\left[\hat a(\vec p),\hat a(\vec q)\right]=\left[\hat a(\vec p)^\dagger,\hat a(\vec q)^\dagger\right]=0\tag{1}$$ $$...
Sirius Black's user avatar
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0 answers
35 views

calculation of partition function: choice of a cross-section

Given a TFT $Z$, I aim to calculate the partition function $Z(\mathbb{S}^2)$ as discussed by Lurie on page-7, Example 1.2.1 in: https://arxiv.org/abs/0905.0465 If I am not wrong I do need to show that ...
Nikhilesh Bairagi's user avatar
4 votes
1 answer
97 views

Hermitian adjoint for 3-vectors; should the energy-momentum 4-vector be written as $P^\mu=\left(p^0,\,p^i\right)$ or $\left(p^0,\,\vec {p}\right)$?

Consider the real Klein-Gordon scalar; Taking the Hermitian Hamiltonian, and (spatial) momentum operators as $$\hat H = \int \frac{d^3 \vec p}{\left(2 \pi\right)^32E(\vec p)}E(\vec p)\hat a^\dagger(\...
FutureCop's user avatar
  • 237
10 votes
3 answers
288 views

How to Fourier transform to find the operator coefficients in the solution to the real Klein-Gordon scalar field in the Heisenberg picture?

From this site for the university of Nottingham QFT notes in the notes for Lecture 3 there is an exercise on page 4 to show that $$\hat a(\boldsymbol{k})=\int d^3x\left[E(\boldsymbol{k})\hat\phi(\...
Electra's user avatar
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1 vote
0 answers
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Existence of operator - From "Symmetry breaking" book by Strocchi

In the book "Symmetry breaking" by Strocchi, in the chapter were he introduces the Fock representations for C*-algebra of bounded operators over a Hilbert space (Part II, chapter 2, page 76),...
MBlrd's user avatar
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0 answers
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On identifying unitarily inequivalent representations of a C*-algebra

Given a finite-dimensional CCR C*-algebra $\mathcal{A}$ (you can find the details of how CCR algebras are introduced in many books, for example see "Operator algebras and quantum statistical ...
MBlrd's user avatar
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0 answers
60 views

Fourier transformation of $\log(q^2)/q^4$ in $d=3$

I currently have the following Fourier transformation that I need to compute \begin{equation} \int \frac{d^3q}{(2\pi)^3}\frac{\log(\mathbf{q^2})}{(\mathbf{q})^4}e^{i\mathbf{q}\cdot \mathbf{r}} \end{...
MathZilla's user avatar
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3 votes
1 answer
120 views

Understanding the Geometry Spawned from Quotient Spaces $GL^+(4,R)/SO(3,1)$, $GL^+(4,R)/Spin(3,1)$, and $GL^+(4,R)/Spin^c(3,1)$

I'm working on a theoretical framework where I explore different quotient spaces formed with GL$^+$(4,R) and various groups. Specifically, I'm interested in the types of geometry that arise from the ...
Anon21's user avatar
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0 answers
62 views

How to go from integral with $\theta((k^+)^2 - \vec{l}_\epsilon^2 - \vec{k}_\perp^2)$ to $\theta(k^+ - |\vec{k}_\perp|)$

I am trying to reproduce the calculation of the so called collinear-soft function, defined in arxiv 1410.6483. More concretely I would like to know the in between steps of the following equation since ...
P Andrea Catalan PA's user avatar
2 votes
0 answers
104 views

Can there be a finite closed form for the one dimensional heat kernel $e^{\frac{d^2}{dx^2}}$ in operator calculus?

In this question we manage to show the existence of a closed form for arbitrary $e^{a(x) \frac{d}{dx} + b(x)I}$ as a single term of the form $k_1(x) f(k_2(x))$ where $k_1, k_2$ obey an interesting ...
Sidharth Ghoshal's user avatar
3 votes
5 answers
260 views

How to give a closed form to $e^{a(x) \frac{d}{dx} + b(x)I}[f]$ in physicists style abuse of notation?

In Quantum Mechanics we have the famous time evolution result (here $a$ is a constant) $$ e^{a \frac{d}{dx}}[f] = f(x+a) $$ Which is an abuse of notation but makes sense due to Taylor's Theorem. In ...
Sidharth Ghoshal's user avatar
0 votes
0 answers
47 views

Tricky "Divergent" Integral: Correction to Groundstate

I am trying to rederive the results presented in the paper, in particular equation (30). That is, I am trying to compute the correction to the ground-state energy of a dipolar condensate due to beyond-...
steveaw123801's user avatar
1 vote
0 answers
96 views

Partition function of a QFT.

There is a YouTube lecture by Robert Dijkgraaf titled:"Introduction to Topological and Conformal Field Theory (1 of 2)." https://www.youtube.com/watch?v=jEEQO-tcyHc&t=2977s At one point ...
Nikhilesh Bairagi's user avatar
2 votes
1 answer
116 views

Can someone solve this integral or what is wrong with it?

I am trying to perform an integral but the main cause of issues and errors is the following part \begin{equation} -\int_0^1dx\: \log(m^2+Q^2x(x-1)) \end{equation} where there are some other $x$'s ...
MathZilla's user avatar
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1 vote
0 answers
54 views

Mellin-Barnes integrals as "master integrals" in code for automated computation of Feynman graphs

I'm writing a code with the aim of automating operator mixing calculations in Euclidean QCD. I have alreaady written a module for generating all graphs and another for generating and assigning Feynman ...
y9QQ's user avatar
  • 99
0 votes
1 answer
94 views

How to diagonalize an infinte dimensional operator

I want to take logarithm of an infinite dimensional operator given by $\rho = \int\int dx_1 dx_1'C(x_1,x_2)C^*(x_1',x_2)|x_1\rangle\langle x_1' |$, where $C(x_1,x_2)$ is a gaussian function in $x_1$ ...
QuantumOscillator's user avatar
8 votes
3 answers
419 views

Show that the real K-G equation, $(\Box + m^2)\phi=0$ is the EOM for the action $S=\frac12\int d^4x(\partial^\mu{\phi}\partial_\mu{\phi}-m^2\phi^2)$

This question concerns a real scalar field. Show that the real Klein-Gordon equation, $(\Box + m^2)\phi=0$ is the equation of motion, $\delta S[\phi(x)]/\delta\phi(x)=0$, for the action $$S=\frac12\...
Electra's user avatar
  • 324
2 votes
1 answer
96 views

The form of the intersection of two von neumann algebras

Let $A$ and $B$ be Type I factor von Neumann algebras of operators on a separable Hilbert space $\mathcal{H}$. Let $B'$ denote the commutant of $B$. Can $A \cap B'$ always be written as a discrete (...
paad89's user avatar
  • 137
1 vote
1 answer
42 views

Do limit and commutator of elements in a quasi-local algebra commute?

I am studying the quasi-local algebra on Bratteli and Robinson Operator Algebras and Quantum Statistical Mechanics, Vol. I (see definition 2.6.3), but there is one thing that is not clear to me at the ...
MBlrd's user avatar
  • 199
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0 answers
53 views

Proof of one of the equations from Schwartz's QFT book

I am attempting to prove an integral that appears in the appendices of the book "Quantum Field Theory and Standard Model," specifically equation (B.25): $$ \dfrac{1}{(2\pi)^{d}}\int\limits_{\...
Jimeens's user avatar
  • 163
1 vote
1 answer
197 views

TQFT vs CQFT vs QFT intro

What is a vague motivational intro to the relationship between topological quantum field theory, cohomological quantum field theory, and quantum field theory? I am a beginner. Here are the vague basic ...
user135743's user avatar
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0 answers
63 views

Any reference for the mathematics of Quantum Mechanics with infinite degrees of freedom?

I am looking for a book, or lecture notes or even courses available on YouTube where there is a good and detailed discussion on the mathematical aspects of Quantum Mechanics with infinite degrees of ...
MBlrd's user avatar
  • 199
2 votes
0 answers
92 views

Series of operators

Given an operator algebra of bounded operators $\mathcal{A}$ acting on a Hilbert space $\mathbb{H}$, I am interested in the algebra of tensor products $\mathcal{A}^{N} = \otimes_{k=1}^{N} \mathcal{A}...
MBlrd's user avatar
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