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.
417
questions
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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{...
- 247
3
votes
1
answer
108
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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 ...
- 2,561
0
votes
0
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56
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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 ...
2
votes
0
answers
79
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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 ...
- 16.6k
2
votes
3
answers
179
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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 ...
- 16.6k
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0
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39
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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-...
1
vote
0
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50
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 ...
2
votes
1
answer
112
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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 ...
- 247
1
vote
0
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35
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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 ...
- 141
0
votes
1
answer
53
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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 $x_1$ ...
7
votes
3
answers
329
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
$$S=\frac12\...
- 316
2
votes
1
answer
63
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The form of the intersection of two von neumann algebras
Let $A$ and $B$ be Type I factor von Neumann algebras of operators on a separable Hilbert space $\mathcal{H}$. Let $B'$ denote the commutant of $B$. Can $A \cap B'$ always be written as a discrete (...
- 117
1
vote
1
answer
27
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Do limit and commutator of elements in a quasi-local algebra commute?
I am studying the quasi-local algebra on Bratteli and Robinson Operator Algebras and Quantum Statistical Mechanics, Vol. I (see definition 2.6.3), but there is one thing that is not clear to me at the ...
- 133
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votes
0
answers
37
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Proof of one of the equations from Schwartz's QFT book
I am attempting to prove an integral that appears in the appendices of the book "Quantum Field Theory and Standard Model," specifically equation (B.25):
$$ \dfrac{1}{(2\pi)^{d}}\int\limits_{\...
- 163
1
vote
1
answer
87
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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 ...
- 371
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votes
0
answers
55
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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 ...
- 133
2
votes
0
answers
79
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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}...
- 133
7
votes
0
answers
344
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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 ...
- 2,108
1
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0
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113
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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 ...
- 789
1
vote
0
answers
36
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]}...
- 81
1
vote
1
answer
83
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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 ...
- 133
1
vote
1
answer
105
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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 ...
- 133
0
votes
0
answers
29
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 ...
- 43
0
votes
1
answer
15
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 ...
- 190
0
votes
0
answers
66
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}+ \...
- 1
0
votes
0
answers
19
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 ...
- 525
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 ...
- 133
2
votes
0
answers
55
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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 ...
- 21
2
votes
1
answer
74
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_{...
- 23
0
votes
0
answers
23
views
Implication of Helmholtz decomposition: Fundamental theorem of vector calculus, for causality in electrodynamics - fixing the unphysical
I've been reading Jackson's Electrodynamics chapter 6.
I want to believe I now understand the fundamental theorem of vector calculus.
A vector field seems to be decomposable into a longitudinal or ...
- 583
0
votes
0
answers
31
views
Evaluation of integral with method of stationary phase
With the integral :
$\int_{-\infty}^{\infty}{dxF(x)\exp{(i\phi(x))}}$
The function $\phi(x)$ is a rapidly-varying function over the range of integration while $F(x)$ is a slowly-varying by comparaison....
- 119
2
votes
0
answers
69
views
How do you calculate the partition function on a manifold-with-corners in extended TQFT?
I'm a physicist trying to study Topological Quantum Field Theory (TQFT), so apologies if the following has some basic mistakes or misuse of terminology. When answering please bear in mind that I'm not ...
- 151
2
votes
1
answer
224
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Witten's theorem on Feynman diagrams
This is a follow-up to a recent question of mine Witten's proof of Wick Formula of QFT. The background of this question can be found there if needed, but I feel my question is simple enough ...
- 708
0
votes
0
answers
60
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$\int d^3x e^{ -i\vec{k}\cdot \vec{x}}\frac{1}{|\vec{x}|} = 4\pi \frac{1}{|\vec{k}|^2}$?
Reading the Schwartz quantum field theory p.235 I came across the following integration (Calculating Born approximation) :
$$ \frac{m_e^2}{4\pi^2}(\int d^{3}x e^{-i \vec{k}\cdot \vec{x}} \frac{e^2}{4\...
- 2,108
0
votes
1
answer
89
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Measuring in Different Bases - Deterministic versus Random outcomes [closed]
The Qiskit Textbook on https://qiskit.org/textbook/ch-states/single-qubit-gates.html in section 4: Digression: Measuring in Different Bases, says –
Z-basis is not intrinsically special, and that ...
- 117
9
votes
2
answers
392
views
Witten's proof of Wick Formula of QFT
Let $\mathcal{S}$ be a finite dimensional real vector space with a positive definite summetric bilinear form $B$. Let $dv$ be a Lebesque measure on $\mathcal{S}$ such that
$$\int_{\mathcal{S}}e^{-B(v,...
- 708
3
votes
1
answer
17
views
Expansion of common expression in Dimensional Regularisation
In QFT we often have it with the expression
$$\frac{\Gamma(\epsilon)}{16\pi^2}\left(\frac{4\pi}{A^2}\right)^\epsilon.$$
This is expandet as
$$\frac{1}{16\pi^2}\left(\frac{1}{\epsilon}+\ln\left(\frac{4\...
- 31
1
vote
1
answer
23
views
Show that all elements in each column of these matrices share the same complex argument
As part of my Final Degree Project I am recreating some of the calculations in this paper by H. Cassini and M. Huerta.
In page 11 we are given the following equations (54-56):
\begin{equation}
\begin{...
1
vote
1
answer
52
views
Wilson Loops in Principal Bundle Language
I'm currently reading Schwartz's book and at page 489 he defines the Wilson Line field and writes in closed-form given by
$$ W_P(x,y)=\exp\left(ie \int\limits^x_y A_\mu(z) dz^\mu\right) $$
I'm looking ...
- 909
1
vote
0
answers
119
views
An expression of certain integral in terms of the Bessel function
Reading Schwartz's quantum field theory book p.220, I came across certain integral : he calculated commutator for a scalar field as
$$[\phi(x) , \phi(y)] = \frac{-i}{2\pi^{2}}\int^{\infty}_{0} q^{2}dq\...
- 2,108
0
votes
0
answers
25
views
Why can we count the number of Nambu-Goldstone bosons by taking the difference between the number of generators $n(G) -n(H)$
I'm taking a break from solving electrodynamics problems in spherical geometries to learn some trivial quantum field theory. I'm reading about spontaneous symmetry breaking.
There is an example in Zee'...
- 583
1
vote
0
answers
26
views
path integral of scalar field action on Riemann surface with boundary
In their paper 'Sewing Polyakov Amplitudes I: Sewing at a Fixed Conformal Structure' Carlip et al. start by considering the path integral $$Z_{\Sigma'}[\tilde{X}] = \int [dX]e^{-S[X]} $$ for the ...
1
vote
1
answer
112
views
Determinant of Integral operators
According to Wikipedia, the Fredholm determinant of $1-T$ with a trace-class integral operator $T$ with kernel $K$ is ("informally") defined by
\begin{equation}
\text{det}_F\big(1-T\big)=\...
- 81
0
votes
0
answers
19
views
Is this the right way to find the size and the angle with respect to the given z-axis for the nitrogen atoms spin?
A nitrogen atom has in its ground state a spin with a quantum number s=3/2:
. Determine the possible results of measurement of the size and the angle
with respect to the given z-axis for the nitrogen ...
- 173
0
votes
0
answers
40
views
Fourier transform and required spectral function to give a Chern-Simons propagator
What (spectral) function $\Psi_{ij}(k)$ is required to give the following Fourier integral
in $\mathbb{R}^{3}$ and how do you prove it? Does it even exist? I'm stuck.
\begin{equation}
\frac{1}{(2\pi)^{...
- 65
0
votes
0
answers
18
views
Understanding the boundary condition of spherical waves in the flat spacetime
I am trying to understand one of the two boundary conditions one has to impose to find the solutions of the wave equation in the flat space-time inside a collapsing null shell. For the spherical wave, ...
0
votes
0
answers
63
views
Singularies of Complex Integrals
My question is about singularities and branch cuts in complex analysis. I found them from studying physics.
I've been reading a book on Quantum Field Theory by Claude Itzykson and Jean-Bernard Zuber. ...
- 487
3
votes
2
answers
189
views
Green's Function and Zero Modes of Differential Operators
I have this question from reading QFT textbooks.
Consider harmonic oscillator as a simpler model for differential operators. The Lagrangian is given by
$$L=\frac{1}{2}\dot{x}^{2}-\frac{1}{2}\omega^{2}...
- 487
1
vote
0
answers
42
views
Problem with the Wightman axiom about the transformation law of the fields
The axiom in the title is almost always stated as
$$
U(a, A)^{\dagger} \phi(x) U(a, A)=S(A) \phi\left(A^{-1}(x-a)\right)
$$
or, more precisely, as
$$
U(a, A) \phi_n(f) U(a, A)^{-1}=\sum_{m=1}^N S\left(...
- 425
1
vote
0
answers
68
views
Making Mathematical Sense of UV-cutoff in QFT
My question is about quantum field theory, but I post it here because I want to understand the mathematics.
In the functional integral approach to quantum field theory, one considers the functional ...
- 487