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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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Rewriting a state as a field in CFT

I've been working through a textbook and course on conformal field theory recently. However in a section illustrating how to calculate correlators for secondary fields (using the free boson as an ...
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28 views

Proving a state is a KMS state

Define an *-algebra generated by the symbols $W(u)$ for $u\in \mathbb{C}$ subject to the relation $$W(u)W(v) := e^{\frac{1}{2}i\Im(\overline{v}u)}W(u+v), W(u)^* = W(-u)\quad u,v \in \mathbb{C}.$$ ...
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25 views

Proofing equation containing time-ordering operator

Preparing for a presentation at university (I'm a Bachelor physics student) I have come across the formula below containg the time-ordering operator $T$. Although i have now understood the action of ...
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0answers
24 views

Navier-Stokes smoothness problem and Gauge Theory

Recently, I came across this paper where the author describes an analogy between electrodynamics and fluid dynamics. He develops a one-to-one correspondence between the equations of electrodynamics ...
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1answer
58 views

Rigorous proof of quantum electrodynamics renormalization

In most physics books they give proofs of renormalization of quantum electrodynamics that are not mathematically rigorous. Is there any book or article that give a formal proof of quantum ...
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13 views

Problem in proof of Wick theorem

In the proof of Wick theorem (https://arxiv.org/abs/math/0406251) the autor for the the set $i_1 ,i_2 ,...,i_m$ of indices define the polynomial $$P(b)_{i_1,...i_m}=(\partial_{i_1}+\sum A^{i_1i}b_i)\...
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1answer
139 views

Getting representations of the Lie group out of representations of its Lie algebra

This is something that is usually done in QFT and that bothers me a lot because it seems to be done without much caution. In QFT when classifying fields one looks for the irreducible representations ...
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2answers
46 views

Going from one notation to another in Yang-Mills

In many books on Yang-Mills theories, written from a physicist's point of view, the curvature tensor is written as: $$F_{\mu \nu}^a = \partial_\mu A_\nu^a-\partial_\nu A_\mu^a+gf^{abc}A_\mu^bA_\nu^c$$...
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1answer
107 views

What constitutes a gauge theory? Help me understand electromagnetism as the prototype of all gauge theories

In The Geometry of Physics and Knots, Sir Michael Atiyah says that electromagnetism is the prototype of all gauge theories: The prototype of all gauge theories is electromagnetism. From the ...
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28 views

Fourier Series and Delta Function

Can you please help me understand what this equation means (and proof), and what exactly $\delta_{k,0}$ means. $$\sum_{k} e^{ik} = N\delta_{k,0}$$ where $N$ is some constant. I found this equation ...
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1answer
41 views

Functional Derivative for Specific Question

Can you help me understanding how author got to equation 1.12, and what is phi(X)function. (https://i.stack.imgur.com/16LOQ.jpg) $$J[f] = \int [f(y)]^p \phi{(y)} d{...
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1answer
183 views

In the Physicists' definition of the path integral, does the result depend on the choice of partitions?

The standard definition of the path integral in Quantum Mechanics usually goes as follows: Let $[a,b]$ be one interval. Let $(P_n)$ be the sequence of partitions of $[a,b]$ given by $$P_n=\{t_0,\dots,...
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0answers
44 views

Knot invariant in arbitrary 3-manifold

In famous Witten's paper "Quantum Field Theory and Jones Polynomial", Witten proposed $\int DA \exp{iL} \prod_{k=1}^{r} W_{R_t} (C_i)$ as the knot invariant in ARBITRARY 3-manifold. ($L$ is a ...
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1answer
82 views

Type III Von Neumann algebras and spectra of the modular operators.

I´m studying the paper of Fredenhagen. There he said that he would prove that the algebra of local observables under certain conditions is of type III, by showing that all modular operators satisfy $...
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47 views

Basis for a tensor product of Fock spaces

Let $H, K$ be two Hilbert spaces with orthonormal bases $\{e_{\alpha}\}, \{f_{\beta}\}$, respectively. Now, I already found a result proving that a basis for the Fermi Fock space generated from $H$, ...
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0answers
58 views

Quantum Groups for Generic q and 3d-TQFT. What breaks?

I've just started looking through Quantum Invariants of Knots and 3-Manifolds by V.G Turaev and want to understand what exactly is breaking in the construction of a 3d-TQFT when one considers the ...
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24 views

Why do the $\Gamma$ matrices behave like vectors and tensors in the spinor representation of SO groups?

One of the things that confuse me most when I study group theory and quantum field theory is that I constantly run into the situations where $\psi C \Gamma_M\chi$ are treated like vectors, $\psi C\...
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1answer
69 views

Left cosets complex orthogonal group under real orthogonal group

In the lecture notes at http://www.math.ias.edu/QFT/fall/lect2.ps (page 2) there is a "standard" lemma: In this lemma $G = \mathrm{SO}(n,\mathbb{R})$ and $G_{\mathbb{C}} = \mathrm{SO}(n,\mathbb{C})$ ...
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0answers
24 views

Skew product of Hilbert Spaces

I’m researching into relations of Fock spaces (in particular so-called “exponential types”) and in the book Introduction to Algebraic and Constructive Quantum Field Theory by Segal, Baez and Zhou they ...
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0answers
36 views

Is the category of $2$-topological quantum field theories locally small?

At first, looking at 2TQFT I see no reason to expect it to be locally small. But we know that 2TQFT is equivalent to the category of commutative Frobenius algebras cFA$_{\mathbb K}$, which is locally ...
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1answer
137 views

Covariant derivative: QFT vs. Math

In class, we have seen that the covariant derivative of some form $R$ can be written as: $$DR = dR + [A, R] = dR + A\wedge R - R\wedge A \tag1$$ Here, $d$ represents the external derivative over ...
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1answer
223 views

How do I prove that the angular momentum is a Hermitian operator?

Confirm that the operator $$\hat I_z= \left(\frac hi\right)\frac{d}{dφ},$$ where $\varphi$ is an angle, is Hermitian.
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86 views

Integrate $ I=\int_0^1 dx\ln\bigg(\frac{A}{A-x(1-x)B}\bigg) $

I am currently working on a mathematical problem in QFT when I came across the integral: $$ I=\int_0^1 dx\ln\bigg(\frac{A}{A-x(1-x)B}\bigg) $$ I have no idea how to do such an integral, or where to ...
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0answers
59 views

Looking for easy materials on Conformal Field Theory for beginners

Question: I'm a math student in senior year. I want to know about CFT(as the title explained). For related knowledge, I've learned basics about Differential Geometry, Differential manifolds. I want to ...
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1answer
49 views

Functional derivative normalization sensitive to normalization of test function?

Background: I am a physicist with decent background in mathematics. Reading the article on functional derivatives on wikipedia gives: The functional derivative $\frac{\delta F}{\delta \rho(x)}$ of ...
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1answer
43 views

Field Theory Phase Factor vs Anomaly

In this paper on topological quantum field theories the authors discuss something called the anomaly in section 5. In Witten's paper on field theory and the Jone's polynomial he discusses something ...
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1answer
52 views

Difference between infinitesimal parameters of Lie algebra and group generators of Lie group

I am getting myself confused regarding the differences between the infinitesimal generators of Lie group and the elements of the Lie algebra, likely due to the fact that I am studying from a physics ...
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0answers
64 views

Pinors vs Spinors

I was reading the paper "The Pin Groups in Physics: C, P, and T" by M. Berg, C. Morette-DeWitt et al. in which they analyze the (double) covering groups of (Lorentzian) orthogonal groups $\...
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0answers
73 views

What is $H_{3}Spin(3)$, and how is this related with the twist of framing on a 3-manifold?

From the question, Mr Ryan Thorngren said in the answer that the the framing anomaly of the gravitational Chern-Simons action $$I(g)=\frac{1}{4\pi}\int_{M}\mathrm{Tr}(\omega\wedge d\omega+\frac{2}{3}...
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0answers
55 views

Solving an integral related to QFT, using identity 3.914 from Gradshteyn-Ryzhik

Given the following formula (#3.914, Gradshteyn-Ryzhik,1980): \begin{equation} \displaystyle\int\limits_0^\infty e^{-\beta{\sqrt{\gamma^2+x^2}}}\cos(bx)dx=\frac{\beta\gamma}{\sqrt{\beta^2+b^2}}K_1(\...
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0answers
48 views

Question on Turaev's paper about axioms for topological quantum field theory

I am currently reading Turaev's paper Axioms for topological quantum field theory. In couple of place, there is a paraphrase "... is natural with respect to $\mathfrak{U}$-homeomorphism" and I don'...
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0answers
40 views

Is the comparator $U(y,x)$ in Gauge Theory the same as a holonomy?

For Gauge theories you have a comparator that transforms as $$U(y,x) = e^{i\alpha(y)}U(y,x) e^{-i\alpha(x)}$$ Is this the same thing as the holonomy?
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11 views

Delta/metric question (context commutator poincare transf.)

The problem statement, all variables and given/known data Relevant equations I believe that $\frac{\partial x^u}{\partial x^p} =\delta ^u_p $ (1) $\implies $ (if $\delta^a_b $ is a tensor, I'm not ...
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1answer
140 views

Functional derivative in QFT

Introductory overview I have that $$iW_0[J] := -\frac{1}{2}\int d^4x d^4 y J(x)D_F(x-y)J(y)$$ and I'm trying to perform the calculation of a two-point function $G^{(2)}(x,y)$ from the fully ...
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1answer
160 views

Integral of $e^{-xy}$ is a Dirac Delta

I was looking at papers about the SYK model page 33 (equation 112), in which they write $$\int\mathscr{D}\Sigma\,\mathscr{D}G~e^{-\frac{N}{2}\int\limits_{\left[0,\beta\right]^2} d\tau_1d\tau_2\Sigma\...
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0answers
43 views

QFT - Generating Functional

The problem statement, all variables and given/known data Hi I am looking at the attached question part c) Relevant equations below The attempt at a solution so if i take $\frac{\partial^{(n-1)}}...
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1answer
98 views

On Inequivalent Representations

Kronz and Lupher in their article, "Unitarily Inequivalent Representations in Algebraic Quantum Theory" say: "Fock representations accommodate systems having infinite degrees of freedom; but they ...
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1answer
362 views

What rigorous mathematical theorems has Edward Witten discovered?

I read that Ed Witten's 1990 Fields Medal was somewhat controversial among mathematicians, because even though no one questioned his deep conceptual understanding of important new mathematical ideas, ...
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1answer
192 views

Peskin & Schroeder equation 2.54 clarification

Let $x^0 > y^0$ where the zero above means the time component of four-vectors. This only means that the points $x,y$ in space-time are not occuring simultaneously. Then the equation is $$\int \...
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72 views

Does the singularities of black holes not being simply connected imply that the universal coverings of quantum field theory fail?

Does the singularities of black holes not being simply connected imply that the universal coverings of the usual types of Lie groups of quantum field theory fail? I was downvoted and told that I need ...
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0answers
42 views

QFT, Noether and Invariance, Complex fields, Equal mass

The problem statement, all variables and given/known data Question attached: Hi I am pretty stuck on part d. I've broken the fields into real and imaginary parts as asked to and tried to compare ...
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0answers
26 views

Feynman Vertex Rule, Correlator, 2 diff coupling constants, why no-cross terms

The problem statement, all variables and given/known data Vertex Feynamnn rule for computing the time correlator of fields under an action such as, for example, Say $S_{int} [\phi] =\int d^4 x \...
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0answers
54 views

Computing energy-momentum tensor with arrowed Feynman slash derivative

Given a four component Dirac spinor $\psi$ (working classically) and the usual Dirac matrices $\gamma^{\mu}$, one can construct a Lagrangian $$\mathcal{L} = \bar{\psi}(i\overset{\leftrightarrow}{{\...
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362 views

How are topological invariants obtained from TQFTs used in practice?

Topological quantum field theories (TQFTs) are studied for different reasons, as exemplified in the following places: Atiyah, Topological quantum field theory Lurie, Topological Quantum Field Theory ...
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1answer
54 views

Spinor chiral transformation by $\psi \to \gamma^5 \psi$

Let $\psi$ be a spinor. Let $\gamma^0,\gamma^1, \gamma^2, \gamma^3$ be the usual gamma matrices and the fifth $\gamma^5 : = i\gamma^0\gamma^1\gamma^2\gamma^3.$ Then if we define $\psi \to \psi' := \...
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0answers
55 views

QFT, more a QM Question, Hamiltonian relation time evolution

The problem statement, all variables and given/known data Question attached here: I am just stuck on the first bit. I have done the second bit and that is fine. This is a quantum field theory course ...
2
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1answer
100 views

Quantum invariants of 2-knots

I'm looking for a status report on analogues of quantum invariants of knots, for the 2-knots (homotopy classes of spheres / other Riemann surfaces embedded into 4-manifolds). Background I'm mostly ...
1
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1answer
354 views

Complex Gaussian integral

I want to calculate this integral $$I=\int \prod\limits_{i=1}^N{d \bar z_{i}dz_{i}\exp[\sum_{i,j=1}^N \bar z_{i}M_{ij}z_{j}+\sum_{i=1}^N(\bar z_{i}f_{i}+\bar g_{i}z_{i}})],$$ where $z_{i}$ are complex ...
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2answers
280 views

Yang–Mills theory and mass gap

I am interested in widening my knowledge into the formal aspects of Yang–Mills theory. In particular, I would like to study the current mathematical and physical research literature about this ...
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1answer
43 views

Conserved currents under Lorentz Transformations

I'm reading David Tong's notes on quantum field theory, and I had a question from page 17 (equations 1.54-55), where he is deriving the conserved currents that arise from a symmetry under a Lorentz ...