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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51 views

Representation of The Poincare Group

I am currently trying to understand the representations of the conformal group. I am following the script by J.D Qualls. At page 29, the author finds the effect of $L_{\mu\nu}$ by "studying the ...
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1answer
80 views

Asymptotic form of a Coulomb-like integral

I need to evaluate or work out the asymptotic scaling of the following integral: \begin{equation} I~=~\int_{\mathbb{R}^3} dq d^2p \frac{e^{i\vec{p}\cdot \vec{r}}e^{iq z}}{p^2 + \frac{1}{g^2}q^4} \end{...
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1answer
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Coulomb matrix element in Quasi-2D bilayer system

In the paper "Electron-hole pairing of Fermi-arc surface states in a Weyl semimetal bilayer" (arxiv) the authors derive an interlayer Coulomb matrix element as (Appendix A). \begin{align} V_{...
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1answer
48 views

Representation of $SO^{+}(3,1)$ for scalar fields

As far as i know, the generators of the representation of the group of the orthochronous Lorentz transformations $SO^{+}(3,1)$ can bewritten in the following form: $$J^{\mu \nu} = i(x^{\mu}\partial^{\...
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22 views

External vs background field

In quantum field theory, what is the difference between an external field and a background field? For instance, a background magnetic field vs an external magnetic field.
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Can the full set of II. Bianchi identities be derived from the symmetries of the action?

In pseudo-Riemannian geometry we can derive the II. Bianchi identities by considering, e.g. the expression of the Riemann tensor in Riemann normal coordinates. They read $$R_{\mu\nu\kappa\lambda;\...
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64 views

Doubt in the definition of the stress-energy tensor in Peskin and Schroeder's QFT book

We can describe the infinitesimal translation $$ x^{\mu} \to x^{\mu}-a^{\mu} $$ alternatively as a transformation of the field configuration $$ \phi(x) \to \phi(x+a) = \phi(x) + a^{\mu} \phi(x). $$ ...
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1answer
50 views

Interaction for QED with charged, scalar particles

Let $\mathcal{L}$ be the Lagrangian for usual QED with scalar, charged particles (with photons and electrons as well): $$\mathcal{L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\bar{\psi}\left(i\gamma^{\mu}\...
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41 views

Space-time Integral of Cubic Field(s)

Context Essentially, I am trying to analytically evaluate $$\tag{1}\int\zeta\phi^2$$ where $\zeta$ and $\phi$ are scalar fields. I have been using various formulae from Mikko Laine and Aleksi Vuorinen'...
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35 views

RG flow of quadrupole coupling in 6+1 dimension electrostatic problem

Given a classical action which describes electrostatics in 6+1 dimensions $$\frac{S_\text{eff}}{T}=\frac{1}{2} \int d^6x(\nabla\phi)^2+\frac{S_{p.p}}{T}$$ where the one particle action, that ...
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2answers
84 views

Why aren't all actions conformally invariant?

I am very confused about coordinate invariance of actions in classical field theories on arbitrary background spacetime or even with dynamical metric. From this question, we see that if the integrated ...
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38 views

Finite norm for solutions of K.G. equation

Before getting into my actual question, let me give an example of a similar problem and its solutions. In non-relativistic wave function quantum mechanics, one usually assigns the Hilbert space of the ...
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40 views

Conformal transformation and coordinate changes

Let $(M,g)$ be a spacetime and $x^\mu$ some local coordinates. We consider a massless scalar field $\phi$ on this fixed spacetime: \begin{align} S[\phi]&=\int_Md^nx\sqrt{-g(x)}~g^{\mu\nu}(x)\...
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What will be the direction of motion of a test charge released at a point in an electrostatic field? [duplicate]

I have recently come across a statement in my book that, "when a test charge is released at a point in an electrostatic field, the direction of traversal of the test charge will not be along the ...
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2answers
87 views

How are scalar fields = particles?

Two such particles I'm thinking of are the inflaton & the Higgs. They are both scalar fields, but they're also both particles with well-defined masses. How is it that scalar fields correspond to ...
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4answers
74 views

Does an electromagnetic gauge transform induce a $U(1)$ transform on the field?

For the free complex scalar Lagrangian, $$\mathscr{L}=\partial_\mu \phi\partial^\mu\phi^{\dagger}-m^2 \phi \phi^{\dagger} $$ if we want it to be invariant under a transformation of the form $\phi\...
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46 views

Mass in Lagrangian, squared or not

How come sometimes we have $m\phi^2$ in a Lagrangian (e.g. Dirac) but we also have $m^2\phi^2$ instead (e.g. scalar field). I was reading about Higgs mechanism, for W/Z boson we have $m^2$ equals ...
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1answer
75 views

Renormalization group theory, first-order phase transitions, perturbative calculations

Consider a satistical field theory that is defined by the generic free energy (or action, in the case of dynamical theories) \begin{equation} F[\phi] = \int_{k,t} a_k |\phi_k|^2 + \mathcal{V}(\phi),...
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Lorentz Transformation of massive Spin-1 fields

First, let me give a summary of the relevant background: In the QFT book by Schwartz, in Chapter 8.2.2, we derive a Lagrangian for a massive Spin-1 field: $$ \mathcal{L}=\frac{1}{2} A_{\mu} \square A_{...
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Gauge orbit orthogonality in a gauged linear sigma model on $\mathbb{C}P^{N}$

I am back with another question from the book Mirror Symmetry, this time from Section $15.1.1$. Consider the gauged linear sigma model for $N$ complex scalar fields and the Lagrangian: $$ L=-\sum_{i=1}...
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37 views

What equation describes the matter field in an electrostatic potential?

I study mathematics at a Russian university, and one of my courses is physics, the course goal being to introduce mathematical methods used in physics to us math students. I got puzzled by one of the ...
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1answer
36 views

Understanding the conformal invariance of the scalar massless wave equation

It can be shown mathematically that the scalar massless wave equation is conformally invariant. However, doing so is rather tedious and muted in terms of physical understanding. As such, is there a ...
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1answer
41 views

Massive Spin-1 Lagrangian: Removal of Spin-0 degree of freedom

I am currently reading Schwartz on QFT and the Standard Model and arrived now at Chapter 8.2.2, where he derives a Lagrangian for a massive Spin-1 field. The final Lagrangian looks like this: $$ \...
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1answer
54 views

Deriving the Generalized Fierz Transformation from Schroeder's Textbook

I am self studying QFT from the textbook An Introduction of Quantum Field Theory and the corresponding solutions from Zhong-Zhi Xianyu. The generalized Fierz Transformation is derived in problem 3.6. ...
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19 views

Gauge theoretic formulation of a classical particle in an electromagnetic field

In gauge theory, we can view the electromagnetic potential as connection on the principal bundle. Wave functions are sections of the associated bundle and we can derive i.e. the Klein-Gordon equation ...
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0answers
47 views

QM is Feynman path ensemble - can QFT be viewed as Feynman field ensemble?

While classical mechanics uses single action optimizing trajectory, QM can be formulated as Feynman ensemble of trajectories. As in derivation of Brownian motion, mathematically it is convenient to ...
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0answers
26 views

How to get the group generators from the coordinate transformation for field?

I'm reading Di Francesco' CFT recently and have some problems with how to find the generator of the conformal group. In (4.15) he showed the conformal transformations in $d\geq3$, and he use $\phi'(x')...
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2answers
62 views

Delta functional representation in response field formalism

A general way of obtaining a field-theoretical description of Langevin dynamics is via the Martin-Siggia-Rose (MSR) response fields. This is essentially just representing the identity - up to some ...
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1answer
23 views

Hermitian conjugate of four derivative $\partial_\mu$

I want to find the hermitian conjugate of $\partial_\mu$ for the real scalar Lagrangian defined as- $\mathcal{L} = \frac{1}{2} (\partial_\mu \phi)^\dagger(\partial^\mu\phi) - \frac{1}{2}m^2(\phi^\...
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1answer
53 views

Klein Gordon equation from its classical Hamiltonian

I don't have much experience in classical field theory and have been trying to study it for the past week. However, I don't know if my understanding of the equations of motion for the fields are ...
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0answers
28 views

Peskin Schroeder 2.44 application [duplicate]

After 2.44, the book calculates the commutators of $\phi$ and $\pi$ with the Hamiltonian. But in the first expression the Hamiltonian has a different form than in the second. The first one matches ...
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1answer
71 views

Is $\phi^4$ theory unstable?

I tried to write a simulation of a $\phi^4$ theory for 2+1 dimensions. But whatever values I gave for the coupling constant it always seems to blow up. i.e. a wave-like equation with a mass and ...
2
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1answer
85 views

Help with a supersymmetry problem 3.5b in Peskin and Schroeder

I am self studying Quantum Field Theory and I am using the book Introduction to Quantum Field Theory by Peskin and Schroeder along with the solution manual by Zhong Zhi Xianyu. I am currently working ...
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1answer
51 views

What is the role of the dilaton in Jackiw-Teitelboim 2D gravity?

I read that the Einstein Hilbert action is topological in 2 dimensions. (What does that mean?). To write down a non-trivial action one introduces the dilaton field in JT gravity. Does this field have ...
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1answer
63 views

Question about Klein-Gordon field Lagrangian

I was studying Klein-Gordon field with Peskin QFT. I know that the Hamiltonian of the scalar field can be written as $$H=\int d^3x\left[\frac{1}{2}\pi^2+\frac{1}{2}(\nabla\phi)^2+\frac{1}{2}m^2\phi^2\...
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2answers
35 views

Notations and Variations in the proof for Noether's theorem for fields

In the proof for Noether's theorem, as given in D.Gross's notes, there are two kind of variations used. $x'^{\mu}=x^{\mu}+X^{\mu}_\alpha(x)\omega^\alpha$ $\phi'_i(x')=\phi_i(x)+\Psi_{i\alpha}(x)\...
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0answers
16 views

Charge inside a hollow dielectric sphere

I'm a bit confused on how to find the displacement field and electric field for a neutral, hollow sphere made from a linear dielectric material. The sphere has inner radius a, outer radius 2a, ...
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0answers
34 views

S-matrix calculation using wave equation with some boundary conditions

Is it possible to map the S-matrix calculation for a Scalar theory into a solving a wave equation with some boundary conditions?
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0answers
41 views

What are the field equations for the particles of the Standard Model?

The field equations for a photon are just Maxwell's equations. But what about all the other particles? From their respective Lagrangians (Klein-Gordon, Proca, Dirac) what are the field equations of ...
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0answers
27 views

Energy-momentum tensor of a fluid for scalar fields

I know that the energy-momentum tensor for a perfect fluid in General Relativity is given by $$T_{\alpha \beta} = (\rho +p)u_{\alpha}u_{\beta}-p\,g_{\alpha \beta}, $$ where $\rho$ is the density, $p$ ...
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0answers
8 views

What is 'negative' magnetic shear? (plasma physics, magnetohydrodynamics, fusion)

While studying plasma physics, and fusion reactor design, I have repeatedly come across 'magnetic shear', simple enough and explained in some depth. But, then, there will sometimes be a brief mention ...
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0answers
13 views

Scaling dimension of non-relativistic field

Let us consider a field theory determined by an action $S$ which depends on a single field $\phi$. In usual relativistic field theories, in natural units, the action is dimensionless, which fixes (via ...
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42 views

Quantum gauge transformation definition

I have always thought that a gauge transformation of a quantum Hamiltonian $H(\Psi,\Psi^{\dagger},A)$ ($A$ is the vector potential and $\Psi$ a matter field) is given by: $$\Psi(r) \rightarrow \Psi(r) ...
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61 views

Why do we study symmetries only via Noether conserved currents?

In general, we say a transformation is a symmetry of a theory if it leaves the action invariant, i.e. if $$S \to S' = S,$$ up to, perhaps, a boundary term (b.t.). However, it is known (see e.g. this ...
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1answer
53 views

Does this Lagrangian density represent anything “real”?

So this lagrangian was used as an example for deriving the equations of motion using the Euler-Lagrange equations in our lecture notes. $$ L (\phi )=-\phi (x,t)^{2}+m\left(\frac{\partial \phi }{\...
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25 views

The correlation length is inversely related to the curvature of the energy density: how?

In the book of Kardar, Statistical Physics of Fields, on page 32, question 1.c it is given that (c) Note that the correlation length $\xi$ is related to the curvature of $\Psi(m)$ at its minimum by $...
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0answers
76 views

The theta term and triviality of principal bundles

Apologies if this question is trivial or has been answered before. If we consider a Yang-Mills theory (with a simple, compact Lie group $G$) on $\mathbb{R}^4$, it is well-known that all the finite-...
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1answer
79 views

Diffeormophism invariance of a non-local series possible?

If I want to construct Lorentz invariant forms involving a scalar field $\phi(x)$ I could have non-local terms such as: $$\int \phi(x)\frac{1}{|x-y|^2}\phi(y) dx^4 dy^4$$ or 'local' forms such as: $$\...
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39 views

Is there a symmetry one can impose to forbid a four point interaction term?

If I have this lagrangian of a real scalar field $S$: $$\mathcal{L}=\frac{1}{2}\partial_\mu S\partial^\mu S-\frac{\mu^2}{2}S^2-\frac{\lambda}{4!} S^4$$ does there exist a symmetry one can impose to ...
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1answer
37 views

How to write lattice $\phi^4$ hamiltonian in terms of Pauli matrices?

I want to decompose lattice~$\phi^4$ hamiltonian in terms of Pauli matrices. Particularly, how can I decompose $$ H_\text{Lattice}=a^d\sum_{{n}\in{Z}}\left[\frac{1}{2}\Pi_{n}^2+\frac{1}{2}\left(\...

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