Questions tagged [field-theory]

For questions where the dynamical variables are fields, that is, functions of several variables (typically, one time coordinate and several space coordinates). Comprises both classical field theory and quantum field theory. Use this tag when the question applies to both classical and quantum phenomena. Otherwise, use the specific tag instead.

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1answer
39 views

How do I obtain the Dirac equation from the Euler-Lagrange equation?

Knowing that the free Dirac Lagrangian is : $$\tag{1} \mathcal{L}= \bar{\psi} (i \gamma^\mu \partial_\mu -m ) \psi$$ and that the Euler-Lagrange equation is: $$\tag{2} \frac{\partial \mathcal{L}}{\...
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2answers
26 views

w do I draw the tree-level Feynman diagram if the interaction term only represents the scalar particles?

Consider the process $$e^+(p_1)+e^−(p_2) \to S(p_3)S^∗(p_4)\tag{1}$$ $S/S^*$ is scalar particle/antiparticle described by the complex scalar field $\phi$ coupled to QED through the Lagrangian: $$\...
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0answers
15 views

How can I simplify the amplitude of the following Feynman diagram as to obtain equation (v)?

I am trying to find the amplitude for the following pair production $$\gamma(p_1) + \gamma (p_2) \to e^- (p_3) + e^+ (p_4)\tag{1}$$ In an old exercise I am given the possible answers but I am not ...
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16 views

How would you define effective values? [closed]

How would you define effective values? For example, effective mass, effective potential, effective Lagrangian, etc...
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28 views

How will we know whether the interacting $\phi^4$ theory eigenstates are bosonic states/symmetric?

For the free scalar field $\phi$, the commutation relation $[\phi(x),\pi(y)]=i\delta(\vec{x}-\vec{y})$ leads to the bosonic commutation relation $[a_p,a_q^\dagger]=\delta(\vec{p}-\vec{q})$. From there,...
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51 views

Confusions on symmetry breaking and classical field theory

I am just reading some material about symmetry breaking and so-called effective action/potential Consider a lagrangian \begin{equation*} \mathcal{L}=\frac{1}{2}(\partial \phi)^2-\frac{1}{2}m^2\phi^2-...
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22 views

Correlation length for ising-Kitaev chain: Coincidence or are they same?

The correlation length for the two-dimensional classical ising model goes as $$\xi_{ising}(T)\sim |T-T_c|^{-\nu};\qquad \nu=1$$ We can map the classical ising model to its quantum cousin, one-...
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1answer
34 views

Deriving the Gauss Constraint in Free Maxwell Theory

In section 6.2 (page 128) of David Tong's Lectures on QFT, Gauss' law is derived for the free Maxwell theory. The result of computing the Hamiltonian of the theory is (eq. 6.17), $$H = \int d^{3} x \...
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0answers
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The generic form of instanton (antiinstanton) in Kahler $\sigma$-model

Suppose we have some compact Riemann surface $\Sigma$ , and scalar field $\phi$, which takes values in some Kahler manifold (target space) $M$. In other words, we have a map: $$ \phi : \Sigma \...
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103 views

Interpretation of different types of transformations on fields in the case of Poincaré group

I have problems understanding how transformations of Lorentz or Poincaré groups act on fields. We can think about two ways of transformation of a field $\phi_r(x)$: $$\phi_r(x) \rightarrow \phi^{'}_r(...
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Number of bosons and fermions in the fundamental irreducible massive multiplet

The fundamental irreducible massive multiplet is given by $$\Omega^{(n)\alpha_1\cdots \alpha_n}_{\;\;\; A_1 \cdots A_n}=\frac{1}{\sqrt{n!}}\left(a^{A_1}_{\alpha_1}\right)^\dagger \cdots \left(a^{A_n}_{...
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3answers
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Are all vector-bosons gauge-bosons?

All QFTs that I come across have vector fields appearing as gauge-bosons. Is there any problem with vector fields that are not gauge-bosons? I am not so concerned about the theory producing results ...
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5answers
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In physics, are all functions fields?

I really confused if there is a function (mostly in physics, functions represents physical quantities) which is not a field? I feel all functions in physics are fields. Is there any functions which ...
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1answer
36 views

First order expansion of Euler-Lagrange equations

I know that in field theory Euler Lagrange equations are $p_i-d_\mu p^\mu_i=0$. (Classical notations, $p_i=\frac{\partial L}{\partial y^i}, p_i^\mu=\frac{\partial L}{\partial y^i_\mu}$). Being a ...
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4answers
94 views

Why fields are solutions of waves equations?

This could be extremely trivial but I am having problems figuring it out. I think I understand properly the difference between waves and fields. A field is a function valued on space or spacetime ...
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2answers
62 views

How is the higgs field excited to give a Higgs boson?

I understand that the excitation of the Higgs field itself is the Higgs boson, and not the Higgs field itself, which does fit somewhat into the little String theory I've read (The excitations of the ...
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1answer
40 views

Rescaling of effective hamiltonian coupling constants in the Wilsonain renormalization group

I am confused about an aspect of coupling constant rescaling in the Wilsonian renormalization group procedure. (I'm following Kardar's "Statistical Physics of Fields, Ch5). I think I understand the ...
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0answers
19 views

How to deal with integral operators in the action, in the path integral of a field theory?

One could imagine adding to the free action of a scalar field theory some non-local operators given as integrals over the base manifold (or over the boundary) of some smooth function of the scalar ...
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1answer
47 views

Perturbative expansion of the S-matrix in QFT

Beginner to QFT - While Taylor expanding the exponential term of the S-matrix, why is the 2nd order term (quadratic time integration term) written with two different dummy variables for time? (...
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1answer
32 views

Is it true that in abelian Chern-Simons theory diffeomorphisms differ from ordinary gauge transformations trivially?

In Henneaux's Lectures on the Antifield BRST Formalism for Gauge Theories, it is claimed in Exercise 1 that diffeomorphisms $\delta_\xi A_\mu=\xi^\rho\partial_\rho A_\mu+\partial_\mu\xi^\rho A_\rho$ ...
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0answers
114 views

Book as supplement to Fradkin's “Field Theories of Condensed Matter Physics”

I am trying to read Fradkin's book "Field Theories of Condensed Matter Physics" but I am finding it to be a bit hard to follow at some places. In particular, I find that Fradkin sometimes throws some ...
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1answer
70 views

What is meant by the “general formula of a scattering process”?

What is meant by the "general formula for the scattering process"? In an old exercise my lecturer gave me, I am told to: Give the Lagrangian for a scalar Yukawa Scattering: $$\mathcal{L}= \frac{1}{...
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0answers
51 views

Lorentz transformation of the spinor fields

I have been reading the Srednicki's QFT textbook (available online at https://web.physics.ucsb.edu/~mark/qft.html) and in Chapter 34 the left and right-handed spinors are discussed. There is a step in ...
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1answer
29 views

Classical chromodynamics force density vector

In classical electrodynamics there is a current and potential four-vector. The covariant force density in a charge continuum according to wikipedia is $f_\alpha=F_{\alpha\beta}J^\beta$ But I'm not ...
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1answer
39 views

What does it mean to have a function of a field?

There was a Leonard Susskind lecture on the Higgs Boson I watched the other day, and he talked about graphing the field where the domain was some sort of field space. The lecture is here at about 6 ...
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0answers
26 views

Invariance of monopole under simultaneous rotations

I'm trying to prove the fact that 't Hooft $SU(2)$ monopole solutions are invariant under simultaneous $SU(2)$ and spatial rotations. I have some idea on how to go about it, but I'm not sure. I went ...
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0answers
12 views

Moduli normalisation

Consider the action for a massless scalar $$ S = \int d^4x\big[\frac{1}{2}(\partial \phi)^2-V(\phi)\big], $$ $V(\phi) = \frac{g^2}{4}(\phi^2-v^2)^2$, which admits domain wall solutions centered at $...
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2answers
80 views

Lagrangian for scalar field in terms of klein Gordon equation

I am Studying Peskin and Schroeder, at page 287 , Lagrangian for scalar field is $$L={1\over 2}(\partial _\mu \phi )^2-{1\over 2}m^2 \phi^2.$$ It can be rewritten as $$L={1\over 2} \phi (-\...
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1answer
54 views

Step on canonical quantization

So I've been trying to solve the expression for the Hamiltonian using the canonical quantization of a complex scalar field and I am not sure of how the following step comes by, from $$\mathcal{H} = \...
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0answers
64 views

Computation of beta function for $N$ scalar fields and introduction of deformations

Consider the Euclidean field theory with $N$ real scalar fields $\phi_{i}$ with Lagrangian density: \begin{equation} L=\frac{1}{2}\partial_{\mu}\phi_{i}\partial^{\mu}\phi_{i}+\frac{1}{2}m^{2}\phi_{...
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0answers
20 views

Chaos and Ergodicity in Hamiltonian Field Theory?

In classical mechanics, one intuitive formulation of chaos/ergodicity (in the loose sense) is that most trajectories should fill up phase space densely over infinite time. A classic example of such a ...
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1answer
63 views

What are some good references for field theory via functional analysis?

Many of the aspects of QFT are traditionally done in ways incompatible with a rigorous mathematical treatment, calling for a variety of tricks to fix essentially what was caused by unjustified ...
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1answer
49 views

What is a “quasi-local” charge?

Could someone please tell me what is a quasi-local charge? For instance, why are Brown-York charges called quasi-local?
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1answer
45 views

Hamiltonian of a quantum field that is minimally coupled to gravity

The action for the gravitational field is known as the Einstein-Hilbert action: $$\begin{equation} S_{G}=\int d^4 x \sqrt{|g|} R \end{equation}$$ where $R$ is the Ricci scalar. The ...
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0answers
37 views

QFT-$\phi^6$ Theory-Counter Terms in 3D [duplicate]

I recently asked a previous question about renoramalisation in $\phi^6$ theory for which there was a great answer but I'm still confused about counter terms in the Lagrangian. I'm mostly confident in $...
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3answers
95 views

Lagrange formalism in field theory

I recently had a discussion with a friend of mine who is like me studying physics. And we might got used to a misconception about the Lagrange-Formalism in field theory. In common field theory books ...
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0answers
33 views

Combinatorics identity for arbitrary value of Spin

I wanted to prove this identity for the general value of $\lambda$ $$ \sum_{n=0}^{\lambda-1} (-1)^n{\lambda-1 \choose n} {\partial^{\left(\lambda-1-n \right)}{\partial_-}^{\left(n \right)}}\left( \...
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1answer
166 views

On TQFT and theories without propagating degrees of freedom

Maybe not a very sensible question, but I would like to know, whether there exist topological field theories (TQFT) with propagating degrees of freedom, or, conversely, theories without propagating ...
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1answer
85 views

Quantum Field Theory - Interacting Scalar Fields

Is there an infinite number of interacting theories? Or is there a limit? For example, I know about $\phi^6$ theory, which is non-renormalisable in 4D spacetime, but I've never really gone beyond $\...
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1answer
79 views

Notation of derivatives in field theory

Some textbooks write $$ \frac{\delta F_{\mu\nu}}{\delta(\partial_\sigma A_\kappa)} $$ which sort of implies the derivative of a functional. Some other textbooks write $$ \frac{\partial F_{\mu\nu}}{\...
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1answer
42 views

On-shell SUSY-transformations for interacting Wess-Zumino model

I'm learning SUSY with Quevedo, Cambridge Lectures on Supersymmetry and Extra Dimensions. Setup: The SUSY transformations of the component fields of a chiral field $\Phi$ are given by (p.41) \begin{...
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1answer
31 views

Equation of motion of massless scalar field in F(L)RW spacetime

Consider the equation of motion of a scalar field $\phi(t,x^i)$, $$\nabla^\mu\nabla_\mu\phi=\frac{dV(\phi)}{d\phi}$$ where $V(\phi)$ is the potential. Specialise to a massless field ($V(\phi)=0$) in ...
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1answer
92 views

Doubt on Action of $\phi^4$ theory

I am reading a paper (arXiv version) in QFT. I am stuck at this point, $$S [\phi (x)]={1\over 2}\int\phi (x_1)D (x_1 -x_2)\phi (x_2)dx_1dx_2 $$ $$+{\lambda \over 4!}\int V (x_1,x_2,x_3,x_4) \phi(x_1)\...
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2answers
229 views

Are there fields, and therefore particles, which do not arise from the quantum harmonic oscillator?

BACKGROUND From what I understand of quantum optics, the creation and annihilation of photons is modeled by a quantum harmonic oscillator. The latter is obtained by applying the quantization "...
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2answers
168 views

Spontaneous symmetry breaking and conservation laws revisited

Crystalline solids spontaneous break the continuous translational and rotational symmetries. According to this lecture by Steven Kivelson, this means that conservation laws such as momentum and ...
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1answer
57 views

Massive spin-1 field and Proca Lagrangian

In his book Quantum Field Theory and the Standard Model, Matthew D. Schwartz derives the Lagrangian for the massive spin 1 field (section 8.2.2). In eq. (8.23) he finds this to be \begin{align} \...
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0answers
40 views

Why is the graviton spin supposed to be 2, and how can it interact with matter? [duplicate]

As far as I know, gravitons are predicted to be spin-2 particles because some extrapolation from GR about non-linearity and antisymmetry. I think I want some clarification on that. What are some ...
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1answer
73 views

Set of zeroes as coset space

I am currently studying Chapter 6 of Coleman S. - Aspects of Symmetry. We study a spontaneously broken gauge theory in two spatial dimensions where the Lagrangian reads: $$ \mathcal{L} = -\frac{1}{4}...
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0answers
14 views

Movement of electrons due to current and electric field

In the channel, electrons move from the source to the right as a current. However, once electrons reach the point where the potential is $V_{DS}(sat)$, it is "swept away by the E-field" to the drain (...
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1answer
41 views

Show that $\Phi^p$ transforms as a scalar primary field

I am trying to solve the following exercise from an old worksheet but I don't even know where to start from: A scalar primary field $\Phi(x)$ of scaling dimension $\Delta$ transforms under special ...

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