The Stack Overflow podcast is back! Listen to an interview with our new CEO.

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.

361 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
15
votes
0answers
456 views

Electric charges on compact four-manifolds

Textbook wisdom in electromagnetism tells you that there is no total electric charge on a compact manifold. For example, consider space-time of the form $\mathbb{R} \times M_3$ where the first factor ...
11
votes
0answers
128 views

Is it known what the necessary and sufficient conditions are for the existence of a “3+1 split” (by means of a foliation) of a (Lorentzian) manifold?

When trying to do physics on a more general pseudo-Riemannian manifold we want to require that there is a foliation of this manifold into three-dimensional subspaces. By this I mean we would like to ...
8
votes
0answers
1k views

Gaussian Integrals : Functional determinant expressed as a trace

Be $A_{ij}$ a symmetric matrix. Then I can easily write $$ \int \exp\left(-\frac{1}{2}\sum_{i,j}x_i A_{ij} x_j+\sum_{i} B_i x_i\right)\; d^nx= \sqrt{(2\pi)^n}\exp\left\{-\frac{1}{2}\mathrm{Tr}\log A\...
7
votes
0answers
191 views

Non-abelian string in QCD?

It is easy to find various/many papers in HEP-lattice talk about "Non abelian string in QCD". What does it mean to say "non abelian string in QCD?" Does "non abelian string" happen for pure Yang-...
7
votes
0answers
207 views

Non-topological solitons in condensed matter physics

As I know most well-known soliton solutions in condensed matter physics are topological ones: kinks, domain walls etc. In field theory there are several examples on non-topological solitons: Q-balls, ...
6
votes
0answers
114 views

Meaning of the simplest potential of quintessence models. Fields in denominator?

I am reading Sec. 1.12 of the Cosmology book by Weinberg. In this section he explains the very simple model of quintessence which attempts to provide a dynamical explanation of the smallness of the ...
6
votes
1answer
406 views

Quantization of free real scalar massless field in 2d

Is there a reference to literature where one explicitly constructs quantization of the free real scalar massless field in the 2-dimensional space-time? In particular, how the propagator looks like? ...
6
votes
0answers
208 views

General Relativity as a Special Relativistic Field Theory

In this question, I want to consider only the classical case. I have seen the statement that general relativity can be considered as a spin-2 field living on a Minkowski background. In that case, you ...
6
votes
0answers
552 views

What is the reason for chiral anomalies in condensed matter systems?

If you consider a massless relativistic fermion theory and you perform a chiral transformation, then you realize that while the classical action remains invariant under this transformation the ...
6
votes
0answers
288 views

One more time about Nordstrom theory

Wikipedia says that Nordstrom theory with equations of motion of the test particle $$\tag{1} \frac{d (\varphi u_{\alpha})}{d \tau} = \partial_{\alpha} \varphi $$ and field equation $$\tag{2} \varphi ...
6
votes
0answers
156 views

The consistency conditions of constrained Hamiltonian systems

I am studying the Hamiltonian description of a constrained system. There are some questions puzzled me for days, which I have been stuck on it. From the lagrangian, we can obtain the primary ...
6
votes
0answers
254 views

Are there known turbulent nonlinear equations where the cascade is a thermal gradient?

In a recent answer (here: The equipartition theorem in momentum space ), I suggested that if you have an appropriate first order equation (in the answer I used a second order equation, but it is more ...
5
votes
0answers
169 views
+50

Magnetic field “lines” in $D \ne 4$ spacetimes

The electromagnetic field is represented by an antisymetric tensor $F_{ab} = -\, F_{ba}$ (the faraday). In $D = 4$ spacetimes, it has 6 independent componenents: 3 describing the electric field: $F_{...
5
votes
1answer
170 views

Lagrangians in field theory and ignorance

The thing that has always bothered me while taking my QFT course was the seemingly arbitrary nature of Lagrangians. For the Klein Gordon equation we just wrote down the simplest Lorentz invariant ...
5
votes
0answers
87 views

Spin-dependence of the directionality of dipole radiation

I am interested in understanding how and whether the transformation properties of a (classical or quantum) field under rotations or boosts relate in a simple way to the directional dependence of the ...
4
votes
0answers
59 views

Lagrangian for system of particles with statistical distribution $f(x_1, …, x_N)$

For system of $N$ particles it is known that it is a good model to take Lagrangian to be (ignoring electromagnetism) $$L = \sum \limits_{i=1}^N \frac{1}{2}(m_i \mathbf{v}_i^2) -U(\mathbf{x}_1, ..., \...
4
votes
1answer
128 views

Connection between gauge invariance and Lorentz invariance

This question is presented in the context of Weinberg's QFT book treatment, in particular considering the electromagnetism chapter. It begins in chapter 5 where Weinberg argues that in order to have ...
4
votes
0answers
64 views

Why is it that the equation of a massless scalar field *must* be conformal invariant?

I'm reading a paper [1], p.111 where it is said that: However, the equation of scalar field with zero mass must be conformal invariant while equation $\square\varphi=0$ does not satisfy this ...
4
votes
0answers
43 views

Lagrangian of Phonon-photon

A quite interesting but also hard problem are Polaritons. As far as I have understand the concept it's about phonons coupling to light. The Lagrangian function should therefore have a term for the ...
4
votes
0answers
59 views

Quantization during phase transition

Consider a scalar field $\phi(t,\vec{x})$ in $\mathbb{R}^{1,3}$ with the following lagrangian $$ \mathcal{L} = \frac{1}{2}\partial_\mu\phi\partial^\mu\phi - V(\phi) $$ where $V(\phi)$ is such that ...
4
votes
1answer
83 views

How groups act on fields in QFT?

I read a lot a posts on how to verify what are the symmetries of a given Lagrangian but I really can't find what I need and can't even get it by myself, this because I don't actually understand how ...
4
votes
0answers
69 views

Decomposition of rank-2 field and local interactions

Any rank-2 tensor can be decomposed in the following way $$ \phi_{\mu\nu} =\phi_{\mu\nu}^{TT} + \partial_{(\mu}\xi_{\nu)} +\frac{1}{4}T_{\mu\nu}s+\frac{1}{4}L_{\mu\nu}(w-3s) $$ where $\phi_{\mu\nu}^{...
4
votes
0answers
276 views

How can I identify gauge transformations of fields with gauge transformations on a principal bundle?

I have some trouble with identifying what we do in physics regarding fields and bundle theory. I start from the following construction which I hope is ok: with a Lie group G and a smooth manifold M I ...
4
votes
0answers
126 views

Instanton contributions in quantum gravity

Suppose a low-energetic System, i.e. a System, where the presence of "classical" gravitational fields can be assumed to be Zero. Classically we would have e.g. the ordinary Minkowski metric or more ...
4
votes
1answer
189 views

Boundary conditions in variational principles

In classical mechanics, the condition to fix the variation of the trajectory at the endpoints has a clear-cut meaning. We want the system to propagate from $x\in\mathcal{C}$ to $y\in\mathcal{C}$, ...
4
votes
0answers
152 views

Running coupling, effective potential and the stability of vacuum

Consider the potential $$V(\phi)=\frac{1}{2}\mu^2\phi^2+\lambda\phi^4$$ where $\phi=\phi(t,\textbf{x})$ is a real scalar field. Let, $\mu^2<0$ and $\lambda>0$ then the potential is bounded from ...
4
votes
0answers
148 views

Is the phrase “coupling constant” interchangable with “ strength of interactions”?

Can I use the terms coupling constant and strength of interactions, interchangeably, or are there more subtleties to the term coupling constant that I am not aware of? Coupling Constants from ...
4
votes
0answers
159 views

Renormalization of Auxiliary Fields

I have the following non-linear sigma model (the base space $\mathcal{M}$ is Euclidean): $$ \mathcal{L}=\dfrac{1}{2\alpha}\int_{\mathcal{M}}\mathrm{d}^2\sigma\ \partial^2X^{\mu}\partial^2X_{\mu} $$ ...
4
votes
0answers
84 views

Moduli space for $CP^N$ and $T^{*} CP^N$ in $\mathcal{N}=2$ SUSY

For complex $\phi$ in $U(1)$ gauge theory, \begin{align} |\phi_1|^2 + |\phi_2|^2 +\cdots |\phi_N|^2 =r \end{align} This equation $|\phi|^2=r$, describes sphere $S^{2N-1}$. Dividing the space of this ...
4
votes
0answers
191 views

Axion Model Field Theory Problem

This is a homework problem for a field theory class dealing with an axion model. Originally, we are given that $$S[a]=\int_Md^4x \frac{1}{2}(\partial_{\mu}a(x))^2$$ has a continuous global ...
4
votes
0answers
133 views

What decides the signs and coefficients of terms in superfield?

I'm working on a problem in 3d field theory and I'm confused about how to write the superfields. Specifically, I'm not sure if the signs and coefficients of terms are purely a matter of convention or ...
4
votes
0answers
494 views

Path integral measure and symmetry

For a generic field theory the path integral measure is defined as, \begin{equation} \mathcal{D}\Phi = \prod_i d\Phi(x_i), \end{equation} where $\Phi$ is a generic field (i.e. it may be scalar, ...
4
votes
0answers
179 views

Asymptotic limit of the two kink solution of the sine-gordon equation

I am reading a paper on the sine-gordon model. The solution for a two kink solution is given as: $$\phi=4\arctan\left(\frac{\sinh\frac{1}{2}(\theta_1-\theta_2)}{(a_{12})^\frac{1}{2}\cosh\frac{1}{2}(\...
3
votes
0answers
59 views

What significance do field-operators have, if they don't correspond to observables because of non-hermicity?

Since field-operators are not always hermitian (for example in case of a complex scalar field, or the dirac-field), they don't (in the quantum-mechanical sense) correspond to observables. Does that ...
3
votes
1answer
51 views

Is there a general behavior of energy gap under renormalization?

Perform real space renormalization on a discrete lattice model with a finite energy gap. Is it always true that under the flow of coarse-graining, the energy gap will only increase? I think the ...
3
votes
1answer
65 views

Why doesn't the Lagrangian depend on higher-order derivatives of position?

This isn't a duplicate of already-answered questions, but rather a follow-up of this answer. The author presents a field-theoretical argument whereby a problematic run-away particle creation is ...
3
votes
0answers
69 views

Confused about the gauge transformation of the amplitude tensor for gravitational waves

Far away from the field sources, where the energy-momentum tensor $$T_{mn}=0 \tag{m,n=0,1,2,3}$$ The linearized EFE becomes $$\Box \bar h_{mn}=0 \tag{1}$$ where $\bar h_{mn}$ is the trace-reverse ...
3
votes
0answers
57 views

What does triviality and non-triviality mean in field theory?

I am studying polymer physics and their basic field theoretical models which has connections to $\Phi ^{4}$ field theory. I frequently come in touch with statements about triviality and non-triviality ...
3
votes
1answer
110 views

Why do we impose de Donder gauge?

In the field language, a massless particle corresponds to irreducible representations of the Lorentz group. In particular, given a spin-2 massless particle, we can embed the creation and annihilation ...
3
votes
2answers
80 views

Chemical Potential and interactions

I'm interested in an model with interactions between different kind's of particles. Each particle species has it's own chemical potential. I want to treat the system in the Matsubara formalism. Here, ...
3
votes
1answer
212 views

Calculation of Matrix Element in “Old-Fashioned Perturbation Theory”

I would like to better under the manipulations/formalism applied in order to evaluate the following matrix element from Schwartz "Quantum Field Theory and the Standard Model" (Eq. 4.16) $$\quad V _ {...
3
votes
0answers
111 views

Can we do a Wick rotation by an angle not being $\pi/2$?

If a state obeys an evolution equation, we can replace t by -t. we get another equation and it is interesting to study its solutions. it we replace t by it (wick rotation) we get again another ...
3
votes
1answer
670 views

Bogoliubov transformation for fermion (exercise in Piers Coleman)

I am trying to solve the exercise 3.2 in Piers Coleman's Introduction to many body physics. It's about fermionic Bogoliubov transformation with only 2 fermion operators $a_{1}^{\dagger}$, $a_{2}^{\...
3
votes
1answer
92 views

Non-linearities in Lagrangian of a scalar field coupled to point-like source

I have an exercise where I did not manage to understand the questions. Basically, I have this Lagrangian \begin{equation} \mathcal{L}=\frac{1}{2}(\partial \pi)^2-\frac{1}{\Lambda^3}(\partial \pi)^2\...
3
votes
1answer
480 views

Variation of scalar field action

I am reading Polchinski's review on AdS/CFT . I have a very simple question, and please help me out. Thanks in advanced. The question is about formula (3.19). The scalar effective bulk action is ...
3
votes
2answers
103 views

Understanding the total spin as the Noether's charge and rotation generator of the Heisenberg model

Consider the Heisenberg model where the Hamiltonian $$H= J\sum_{\langle i,j\rangle}\textbf{s}_i\cdot \textbf{s}_j$$ has continuous rotational symmetry. Since $\textbf{s}_i\in\mathbb{R}^3$, the ...
3
votes
1answer
104 views

Lorentz Symmetry Group as continuous symmetry for limit of discrete spacetime

There is a variety of models of quantum field theory, where discrete spacetime is used as technical support, or even suggested as physical reality. As far as I know, all of such models faced serious ...
3
votes
0answers
338 views

Breakdown of the Legendre transform for the complex scalar field

Suppose we wish to obtain the energy density of the free complex scalar field $\varphi$ as a Legendre transform of the corresponding action. From Wikipedia, writing the action of a free complex scalar ...
3
votes
0answers
278 views

Four vector potential and discrete parity transformations

I am having trouble understanding the effect of parity transformations on the four-vector gauge field (for example). I am working in three dimensions, but the analysis is probably not that different ...
3
votes
0answers
332 views

What is the real space causal Green's function for the Klein-Gordon equation in 5 dimensions?

I want to solve the Klein-Gordon Green's function equation $$\left[\partial_\mu\partial^\mu + m^2\right]G(x, x') = \delta(x - x') $$ in 5 space-time dimensions where the boundary conditions on $G(x,x')...