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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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
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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
39 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
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Correspondence between Feynman diagrams in the correlation (n-point) function expansions for 2 different cutoffs [migrated]

My understanding of QFT is quite elementary. I'm reading through Kevin Costello's book on Renormalization and effective field theory, which is based on Wilsonian low energy theory. The integral for an ...
Yashasvi Aulak's user avatar
3 votes
1 answer
184 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
110 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
2 votes
0 answers
90 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 ...
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References for closed string collisions on finite number of fixed type mutually intersecting orbifolds with D-branes at the orbifold singularities [migrated]

I don't know if this topic is studied at all but I'm looking for good references for closed string collisions on a finite number of fixed type mutually intersecting football orbifolds with D-branes at ...
John Zimmerman'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 $\psi$ ...
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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
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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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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}...
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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 ...
Luthier415Hz'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
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2 votes
0 answers
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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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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
86 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
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9 votes
3 answers
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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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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),...
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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 ...
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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
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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 ...
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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
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0 answers
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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
2 votes
3 answers
193 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
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0 answers
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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
74 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
113 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
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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
  • 141
0 votes
1 answer
70 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
7 votes
3 answers
369 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
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2 votes
1 answer
76 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
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1 vote
1 answer
33 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
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0 answers
42 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
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1 vote
1 answer
123 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
60 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
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2 votes
0 answers
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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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6 votes
0 answers
426 views

Can anyone help to perform this heinous integral? (Peskin & Schroeder's Quantum field theory (4.76)-(4.78)) (Including my own trial )

I am reading the Peskin & Schroeder's An introduction to Quantum field theory, p.105~p.106 (Construction of a cross-section from the invariant matrix element (p.104) ) and stuck at understanding ...
Plantation's user avatar
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1 vote
0 answers
120 views

Mathematical Method of Geometric Second Quantization?

I have recently been studying the method of geometric quantization, and I noticed a few methods in it that seem like they could be used to create a geometric second quantization (specifically of the ...
moboDawn_φ's user avatar
1 vote
0 answers
37 views

Smeared polynomials of creation and annihilation operators

Given a massive free scalar field. We can define the quantum *-algebra of observables as a subset $\mathcal{E}\subseteq C^{\infty}(\mathcal{F})$ given by $$ \mathcal{E}:= \lbrace 1,a[f],\overline{a[g]}...
Hey's user avatar
  • 91
1 vote
1 answer
100 views

Von Neumann algebra decomposition as integral of factors and mixed state decomposition as sum of irreducible states

My question is the following. It is known that any Von Neumann algebra can be uniquely decomposed as integral over algebra factors. It is also know that any mixed state can be uniquely expressed as ...
MBlrd's user avatar
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1 vote
1 answer
115 views

Doubt on Von Neumann algebra decomposition as integral of factors [closed]

I am trying to understand the Von Neumann decomposition, according to which every Von Neumann Algebra can be uniquely decomposed as integral (or direct sum) of factors. More specifically, I am trying ...
MBlrd's user avatar
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0 answers
32 views

An integral involving products of random matrices

I came across an integral involving a product of random matrices that I'm unable to evaluate, and I'm curious about whether or not there's a known solution. For context, I started with the following ...
artag's user avatar
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0 votes
1 answer
19 views

Regularized loop integral confusion when changing integration variable from $l_E\to l^2_E$

I'm currently reading lecture notes on Dimensional Regularization where Wick Rotations are being employed. I understand how to regularize and compute these loop integrals but I'm having trouble ...
aygx's user avatar
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0 answers
83 views

The 2D Ising model relationship with the transverse ising chain

I'm studying the 2D Ising model and saw a solution, that first formalized it by using a transfer matrix $T$ and then by the identification $T = e^{-\hat{H}\Delta t}$ as $H = -\sum_a \sigma_a^{x}+ \...
JCX's user avatar
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0 answers
21 views

Question about the toric code

I am reading this paper on toric code for my research and has some questions about it. $\sigma_j^Z$ appears in the equation for $A_S$ (last but five line) and in the last but four line, it’s defined ...
Coco's user avatar
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3 votes
0 answers
100 views

Properties of reducible representations

I have the following doubt. Let's assume we have two mixed states $\rho_1 = \Sigma_i a_i \omega_i^{1}$ and $\rho_2 = \Sigma_i b_i \omega_i^{2}$ on the same algebra, where the states $\omega$ are all ...
MBlrd's user avatar
  • 165
2 votes
0 answers
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Generalisation of Bloch-sphere rotations to higher dimensional Bloch-spheres

Background Our question concerns the generalisation of Bloch sphere rotations to higher-dimensional Bloch spheres. We note the connection between states of a Hilbert space represented on $S^2$ with ...
fintallrik's user avatar
2 votes
1 answer
94 views

Is Fock space a symmetric/exterior algebra?

Wikipedia's Fock Space entry says that the Fock space is the direct sum of tensor products of $H$. However, it is not represented as $\bigoplus_{n=0}^{\infty} H^{\otimes n}$, but rather as $\bigoplus_{...
Godfly666's user avatar

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