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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14 views

Electric field due to multiple dielectric layers

Multiple questions came across me when I suddenly "invented" the following problem. Suppose we have two point charges $q_1$ and $q_2$ such that the straight line joining them can be divided ...
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Bosonization and supersymmetry

In 2D (time + space) there is no notion of statistic. So particles can be described in terms of bosonic and fermionic fields. Well-known example is Thirring/Sine-Gordon duality. There are also some ...
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Field configuration minimizing the Higgs potential

Why is it that the field configuration that minimizes the Higgs potential is said to be a constant one (meaning $\phi(x) = \phi_0$)? Is it a physical reasoning, which kind of makes sense, since a non ...
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Second quantization, Schrödinger field

Consider a system with eigenstates $|q_i\rangle, i=1,2,\ldots$ Now if we have two indistinguishable bosons, one in the normalized state $|\chi\rangle$ , and the other at the state $|\rho\rangle$, \...
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Physical intuition for spatially constant motion in the XY-model in 2+1D

The XY-model on a 2-torus ($L_1,L_2$) has a lagrangian given by $$ L_{XY}[\theta] = \int d^2 x \frac{\chi}{2}\big{(}\dot{\theta}^2 - (\partial_x \theta)^2\big{)} $$ Fourier expanding $\theta$ as $$ \...
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1answer
35 views

Must a field approach one of its vacua to have finite energy?

I'm reading these Cornell lectures on solitons (link doesn't work right now, but it just worked yesterday), and I can't seem to prove what I thought would be a simple analysis exercise. Namely, ...
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Mass terms for scalar lagrangians?

First off, a pre-question: if I got this wrong, then probably the whole reasoning is wrong as well. Studying the lagrangian for a two-particle scalar field with a quartic interaction in the context of ...
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Time-reversal transformation in Landau & Lifshitz compared to newer sources

To my astonishment I found that the transformation behaviour of scalar fields under time-reversal $T$ in newer sources like Peskin & Schroeder and Srednicki is different from the volume series on ...
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31 views

Functional variation problem in Classical Field Theory (Non Relativistic) [closed]

An exercise of my Homework sheet make a statement about rotational variation on a scalar field $\phi(x)$:\ "Consider a scalar field $\phi(x,t)$ in a lagrangian $\mathcal{L}(\phi, \partial_t \phi, ...
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2answers
45 views

Divergence of the magnetic field $H$

it is known (although I have not found much information about it on books and websites) that, while the divergence of $B$ is always zero ($\nabla\cdot B = 0$), we cannot say the same about $H$: the ...
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1answer
35 views

Electrostatics and Magnetostatics as Field Theories ( isn't the Coloumb law depicting an instantaneous action at a distance)

Take the formula for Coloumb's law. It does not show in any case as to how the static field propagates. What I can feel is that the field of a static charge is ever prevading from the time the charge ...
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Physical Interpretation of Partition Function with Background Field

Integrals of the form $$\langle{\phi(x_1)\cdots\phi(x_n)}\rangle=\frac{1}{Z}\int\mathcal{D}\phi\,e^{-\frac{1}{\hbar}S(\phi)}\phi(x_1)\cdots\phi(x_n)$$ can be evaluated by considering Feynman diagrams ...
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Electric field at any point [closed]

Consider a uniformly charged semi infinite hollow cylinder of radius R and consider any general point at the end face of the hollow cylinder , now we have to prove that the electric field at this ...
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1answer
59 views

Mass and breaking in SSB

I actually have two different sub-questions, both based on the understanding of the quadratic term of the lagrangian, so the answer is probably linked. I will use the example of the linear-$\sigma$ ...
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1answer
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How there can be an explicit coordinate dependence on the Lagrangian, if this arises from a Lagrangian density?

I have a very simple question, but strangely I cannot find any answer on the internet; maybe the answer is too simple that I don't notice. I go straight to the point: if I define a Lagrangian from a ...
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Natural extension of the line element for the Lagrangian to fields over 3+1 spacetime?

We are all familiar with the standard construction of the Lagrangian of a free relativistic particle: Start with the definition $$ S=\int_{\Delta t}ds. \tag{1} $$ Then use the definition of the line ...
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Open gauge algebras apart from supergravity theories

Does anyone know of a gauge system that is not a model of (super-)gravity where the gauge algebra fails to close off-shell?
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112 views

Transformation of Lagrangian and action

Consider the Lagrangian $L(q_i,\dot{q_i},t)$ for $i=1,2, ...n$. Transform (invertibly) $q_i$ to another set of generalized coordinates $s_i=s_i(q_j,t)$. Now, in a different scenario, consider ...
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Why would one want to study the geometry of Lie groups?

Lie groups are commonly used in theoretical physics and mathematical physics. They are useful tools to study simple systems such as the harmonic oscillator. They are also crucial in representation ...
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Why projection operator is not equal to zero, as we can write 1st term as 2nd term or vice versa via raising or lowering index with metric?

$$k^2g^{\mu\nu}-k^\mu k^\nu=k^2P^{\mu\nu}(k)$$ Here 1st term can be written as 2nd term via breaking square term and then raising index.
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Fields that lend themselves to variational principles? [duplicate]

In physics, we often describe the dynamic properties of fields using variational principles like defining an action or a Lagrangian. A field however is simply some function of space $\phi(x)$ so I ...
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2answers
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Hamiltonian formalism of the massive vector field

I am currently working through a problem concerning the massive vector field. Amongst other things I have already calculated the equations of motion from the Lagrangian density $$\mathcal{L} = - \frac{...
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Is the Landau free energy equivalent to the effective potential in QFT?

Is the Landau free energy equivalent to the effective action in QFT? This is some sort of part II question to my previous question on MO.
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102 views

Spin 3 vs spin 2 vs spin 1

I wanna to understand, why when one gonna to construct interacting theory of spin 3, one need also include infinite tower of spins 4, 5, 6 , ... As I know, this statement correct even in classical ...
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27 views

Proton decay through a $(3,1)^{-1/3}$ complex scalar

Suppose we add to the Yukawa sector of the SM a complex scalar $T^\alpha(3,1)^{-1/3}$, where $\alpha=1,2,3$ is a $SU(3)_C$ index and the charges assignment means that it transforms as a triplet under $...
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Is it possible to have a compact abelian $U(1)$ lattice gauge theory on a non-compact manifold?

We have a compact lattice gauge theory if we let $A_{i}(n)\in[-\pi,\pi]$, and if we identify $A_{i}(n)\sim A_{i}+2\pi$. A simple lattice gauge theory in 2+1D then has an action $$S=\sum_{x}1-\cos(F_{\...
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Instantons in 1+1 dimensional Abelian Higgs model

Let's consider the Abelian Higgs model in 1+1 dimensions in Euclidean space-time: $$L_E=\frac{1}{4e^2}F_{\mu\nu}F_{\mu\nu}+D_\mu\phi^\dagger D_\mu\phi+ \frac{e^2}{4}(|\phi|^2-\zeta)^2$$ where $\zeta&...
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Auxiliary fields in non-supersymmetric theory

It is well-known, that in superspace formulation of supersymmetric theories auxiliary fields appear. In present of such fields SUSY transformations are linear and independent of model. Are some non-...
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44 views

A theory where it's convenient to express a scalar quantity as a divergence of a vector

The equations of electrostatics,$$\nabla \cdot \vec{E} = \rho, \quad \nabla \times \vec{E} = 0$$make it possible to introduce the scalar potential $\vec{E} = - \nabla \phi$, which then satisfies the ...
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1answer
35 views

From spins to fields

In statistical field theory, one usually considers the so-called Landau Hamiltonian: $$\beta H = \int d^{d}x\bigg{[}\frac{t}{2}m^{2}(x) + \alpha m^{4}(x)+\frac{\beta}{2}(\nabla m)^{2}+\cdots+ \vec{h}\...
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How does one determine the $R$-symmetry group?

As far as I understand it, the $R$-symmetry group is just the largest subgroup of the automorphism group of the supersymmetry (SUSY) algebra which commutes with the Lorentz group. I know for $\mathcal{...
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1answer
32 views

Electroweak scalar triplet coupled to Higgs

Suppose we add to the SM the following electroweak scalar triplet with hypercharge $Y_T=-1$ $$T=\begin{pmatrix} t^0 & t^-/\sqrt{2} \\ t^-\sqrt{2} & t^{--} \end{pmatrix}$$ where the ...
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40 views

Consistency of transformation of scalar fields with mathematical definition of a representation of Lie algebra and Lie group

Transformations of scalar fields under a Lorentz group transformation are generated by differential operators $L_{\mu\nu}=x_\mu\partial_\nu-x_\nu\partial_\mu$. On the other hand, a representation of a ...
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1answer
72 views

Two-field Symmetry Breaking unitary gauge

Let's consider the following theory: $$L= -\frac{1}{4}F_{\mu \nu}F^{\mu\nu} +{1\over 2} |D_\mu \Phi|^2 +{1\over 2}|D_\mu \chi|^2 + \lambda_1\bigl(|\Phi|^2-\frac{v_1^2}{2}\bigr) +\lambda_2\bigl(|\chi|^...
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1answer
63 views

Are the $\alpha$ and $\beta$ matrices of the Dirac equation unique?

Assume we are dealing wth three spatial dimensions $d=3$ which requires 3 $\alpha$ matrices. Furthermore assume that we are looking for them in the space of 4-dimensional matrices, not in higher ...
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Is the number of spin states necessary in the density of states function?

I'm studying how to calculate the density of states in the final configuration in order to apply Fermi golden rule. For free EM field the following expression is the starting point: $$d^3n=\frac V {(2\...
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1answer
118 views

What is the reason for turning global symmetries into local symmetries?

For example a simple complex scalar field theory has a global $ U(1) $ symmetry where the field $ \psi $ can be replaced by $ e^{ i \alpha } \psi $, where $ \alpha $ is just some real constant, ...
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1answer
87 views

Explicit expansion of the term $\overline{\psi}(i \gamma_\mu \partial^\mu-m) \psi$

Explicit expansion of the term $\overline{\psi}(i \gamma_\mu \partial^\mu-m) \psi$ In QED, one finds the first part of the Lagrangian density to be $\mathcal{L}=\overline{\psi}(i \gamma_\mu \partial^\...
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2answers
74 views

Why is it problematic to regard the Lorentz group as ${\rm SO}(4, \mathbb{C})$? [duplicate]

If the four-vector $x^\mu$ is defined as $x^\mu\equiv(ict,{\bf x})$, instead of $x^\mu\equiv (ct,{\bf x})$, the Lorentz group will be the compact(?) ${\rm SO}(4, \mathbb{C})$ group. But the Lorentz ...
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44 views

Lattice differentiation and Locality

Assume we define the locality of a theory in the following way: Assume we have a theory of real scalars, so this theory is non local if the action has terms like $$\int d^dx\,\phi(x)V(x-y)\phi(y).$$ ...
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1answer
38 views

Derivation of variation

$$\delta S=2\int dx^+dx^-\left(\frac{\partial\delta \phi}{\partial x^+}\frac{\partial \phi}{\partial x^-}+\frac{\partial \phi}{\partial x^+}\frac{\partial\delta \phi}{\partial x^-}\right)=-4\int dx^+...
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1answer
52 views

Lorentz transformation of Weyl fields

In the Srednicki's textbook, Chapter 35, the author states (Equation 35.28): $$ U(\Lambda)^{-1}[\psi^\dagger \bar\sigma^\mu \chi ] U(\Lambda) = \Lambda^\mu_{\,\,\nu} [\psi^\dagger \bar\sigma^\nu \chi ]...
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1answer
39 views

Time derivative of a 4-derivative of a scalar field

Let us consider Lagrangian $$ \mathcal{L} = \frac{1}{2} \partial_\mu \phi \partial^\mu \phi - \frac{1}{2} m^2 \phi^2 $$ with $\phi$ being a scalar field, and Minkowski signature $(+,-,-,-)$. My ...
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Why $\delta (\partial_{\mu} \phi_{i}) = \partial_{\mu} (\delta \phi_{i})$ on field theory?

While one are using Hamilton Principle on density Lagrangian, he may find this $$\delta (\partial_{\mu} \phi_{i}) = \partial_{\mu} (\delta \phi_{i})$$ Why this statement are true?
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1answer
56 views

Better understanding of Gauge-Invariance

This is a long question and the main points are emphasised in bold. Consider a non-Abelian SU(N) gauge theory. $t_a $ is an Hermitian generators of SU(N) so that $$U = e^{i\alpha^a(x)t^a} \tag{1}$$ is ...
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1answer
160 views

How do we know from representation theory that a massless spin-1 particle has only two polarizations?

In chapter 8.2.3 of Schwartz' textbook "Quantum Field Theory and the Standard Model", the author states the following, Finally, we expect from representation theory that there should only be two ...
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1answer
59 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
31 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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31 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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55 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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