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2
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0answers
315 views

Product of $\gamma^5 \sigma^{\mu\nu}$

I'm trying to prove that $\gamma^5 \sigma^{\mu\nu}=\frac{i}{2}\epsilon^{\mu\nu\alpha\beta}\sigma_{\alpha\beta}$ I started with the left hand side and expanded the $\gamma^5$ to ...
1
vote
1answer
47 views

Circulation of the gauge potential around an infinitesimal loop: how to get the correct gauge field strength tensor

I've been puzzling with the problem below for more than a hour since it is misleadingly discussed in some textbooks, so I believe it deserves a solution here. Any comments are welcome. I'm trying to ...
6
votes
1answer
112 views

In which field theories with fermions do string- and fivebrane structures not come up?

A year ago, username @Greg Graviton asked in a thread here about the Spin group as covering of the spatial rotations. A subquestion was: What other groups, even larger than SU(2) are there that ...
1
vote
2answers
188 views

Classical point particles to classical fields

I often hear that in the continuum limit we can study large numbers of particles as fields. I always imagined that by removing all bounds on the number of particles (while keeping total energy, ...
4
votes
0answers
107 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 ...
5
votes
1answer
307 views

Quantum Field Theory: why fields are equal to zero on the boundary?

One of the first assumptions, when introducing the Lagrangian and Hamiltonian in an undergraduate course on QFT is $$ \phi(x)=0\,\text{on the boundary} $$ and this is widely used in many situations ...
2
votes
0answers
106 views

Half-integer Spin and “natural conformal dimension”

If we consider a classical field theory for a massless particle of integer spin $s$, in a curved space-time, one finds that it is "naturally" conformal in a space-time of dimension $2+2s$ For ...
0
votes
0answers
67 views

Explanation of the classical coupling of the Higgs Field to Electromagnetism

I'm interested in learning about the classical coupling of the Higgs Field to Electromagnetism. There are numerous sources explaining the Higgs mechanism quantum mechanically, i.e. How does the Higgs ...
1
vote
1answer
57 views

Integrating out fields from classical systems

Has anyone ever heard of integrating out fields from classical Lagrangians if they are quadratic?
1
vote
1answer
72 views

Gradient of a two-component field

I have a two-component field: $$\phi(\vec{x}) = \left( \begin{array}{c} \phi_1(\vec{x}) \\ \phi_2(\vec{x}) \end{array} \right)$$ with $\phi^T = (\phi_1, \phi_2)$. And I am trying to evaluate: ...
1
vote
1answer
203 views

Are multipole fields, multipole expansion, and multipole radiation the same thing?

Interaction between electromagnetic radiation and nuclei can be written in terms of multipole radiation. Are multipole fields, multipole expansion and multipole radiation the same thing? I have found ...
6
votes
2answers
223 views

How does a SCFT avoid the Haag-Lopuszanski-Sohnius theorem?

According to the Haag-Lopuszanski-Sohnius theorem the most general symmetry that a consistent 4 dimensional field theory can enjoy is supersymmery, seen as an extension of Poincarè symmetry, in direct ...
1
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1answer
54 views

The speed of light and fields

Does the speed of light apply to the speed of (waves?) in every known field, or does it only apply to the electromagnetic field?
4
votes
1answer
606 views

What exactly is the connection between the Jacobi and Bianchi identities

While reviewing some basic field theory, I once again encountered the Bianchi identity (in the context of electromagnetism). It can be written as $$\partial_{[\lambda}\partial_{[\mu}A_{\nu]]}=0$$ ...
2
votes
1answer
158 views

Book Recommendation- Classical Relativistic Fields

My bare bookshelves are crying out for the addition of a new family member, more specifically a book: Discussing the classical Klein-Gordon field, spinor fields, gauge fields and all other matter ...
2
votes
1answer
532 views

Tracelessness of energy-momentum tensor and massless photons

I have read the statement that the tracelessness of the energy-momentum tensor is demanded by the condition of photons being massless. I see how this comes about starting from the canonical ...
9
votes
4answers
492 views

Is the Lagrangian of a quantum field really a 'functional'?

Weinberg says, page 299, The quantum theory of fields, Vol 1, that The Lagrangian is, in general, a functional $L[\Psi(t),\dot{\Psi}(t)$], of a set of generic fields $\Psi[x,t]$ and their time ...
2
votes
1answer
299 views

Primary constraints for Hamiltonian field theories

I am currently trying to carry out the construction of the generalised Hamiltonian, constraints and constraint algebra, etc for a particular field theory following the procedure in Dirac's "Lectures ...
8
votes
1answer
204 views

Boundary currents for Asymptotic Symmetry Group (ASG)

In the context of asymptotic symmetry groups, what is a boundary current? Why is it called a "current"? Context: I'm reading Strominger's recent paper on Asymptotic symmetry group of Yang-Mills ...
7
votes
1answer
570 views

Is Bose-Einstein condensate a good example of a classical massive boson field?

Physically, we know that a BEC has formed if a macroscopic number of bosons occupy a single quantum state. The wave-function $\Psi(x)$ of the latter, normalized to the total number of condensed atoms ...
8
votes
1answer
1k views

Trick for deriving the stress tensor in any theory

In D. Tong's notes on string theory (pdf) section 4.1.1 he explains a trick for deriving the stress-energy tensor which arises from translations in the base manifold of the field theory (in this case ...
8
votes
3answers
768 views

Why is the Yang-Mills gauge group assumed compact and semi-simple?

What is the motivation for including the compactness and semi-simplicity assumptions on the groups that one gauges to obtain Yang-Mills theories? I'd think that these hypotheses lead to physically ...
4
votes
2answers
920 views

Coordinate Transformation of Scalar Fields in QFT

By definition scalar fields are independent of coordinate system, thus I would expect a scalar field $\psi [x]$ would not change under the transformation $x^\mu \to x^\mu + \epsilon^\mu $. Correct? ...
8
votes
1answer
277 views

What are the details of the renormalization of Chern-Simons theory?

What is a good, simple argument as to why Chern-Simons theory' is renormalisable? Any good books/references dealing with this effectively? Why does the $\beta$-function vanish? Thanks!
1
vote
1answer
984 views

Deriving the Hamiltonian density for a free scalar field

I'm working through my old notes on QFT (cf. Ref 1) and I'm not quite sure how to approach the derivation of the Hamiltonian density for a free scalar field (question 2.3 on page 19) and the ...
1
vote
1answer
143 views

Dimension analysis in Derrick theorem

The following image is taken from p. 85 in the textbook Topological Solitons by N. Manton and P.M. Sutcliffe: What I don't understand from the above statement: why $e(\mu)$ has minimum ...
0
votes
0answers
59 views

Physical and dynamical components the four potential

I have a question regarding the four-potential and its gauge symmetry. We have a gauge freedom: $A_{\mu} \rightarrow A_{\mu} + \partial_{\mu}\chi$ Such a transformation does not alter the EM field. ...
0
votes
1answer
95 views

Definition: Coupling [closed]

What does it mean to say that 2 fields are coupled? More generally, what does "coupling" mean?
2
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0answers
140 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, ...
6
votes
1answer
213 views

Proof that we can always find a gauge transformation such that $A_0=0$?

I'm trying to follow Coleman's proof from his lectures "Aspects of Symmetry" on page 200-201. He proofs it is always possible to work in the temporal gauge for a general Yang-Mills-Higgs theory. I ...
3
votes
0answers
493 views

The connection between classical and quantum spins

I have two questions, which are connected with each other. The first question. In a classical relativistic (SRT) case for one particle can be defined (in a reason of "antisymmetric" nature of ...
4
votes
3answers
268 views

Are Field Lines an accurate depiction of reality?

Field lines are used for explaining a wide variety of phenomenon. But is it really an accurate depiction of reality? Is it more accurate to imagine a field in a different manner. For instance, using ...
3
votes
2answers
325 views

Higher order covariant Lagrangian

I'm in search of examples of Lagrangian, which are at least second order in the derivatives and are covariant, preferable for field theories. Up to now I could only find first-order (such at ...
6
votes
2answers
1k views

What does a Field Theory mean?

What exactly is a field theory? How do we classify theories as field theories and non field theories? EDIT: After reading the answers I am under the impression that almost every theory is a ...
1
vote
0answers
86 views

Dimension dependence

The question is related to this one The time averaged total energy, $\bar E$, has the following $\varepsilon$ expansion in $D$ dimension: \begin{equation} ...
1
vote
1answer
150 views

Finding out Energy value

A Lagrangian is given by, $$L= \left(\frac{\pi}{2}\right)^2 R^d \left[\frac{1}{2}\dot A^2 - V(A_{max})\right]$$ $$E=\left(\frac{\pi}{2}\right)^2R^d V(A_{max}) $$ where V (A) now includes nonlinear ...
3
votes
0answers
113 views

Is a solution to the Klein-Gordon equation homeomorphic (or even diffeomorphic) to a solution of an equation with a different covariance group?

Consider some solution $\psi(x,t)$ to the linear Klein-Gordon equation: $-\partial^2_t \psi + \nabla^2 \psi = m^2 \psi$. Up to homeomorphism, can $\psi$ serve as a solution to some other equation ...
3
votes
3answers
919 views

Massless limit of the Klein-Gordon propagator

I am working with the propagator associated to the Klein-Gordon equation, as derived in "Quantum Physics a functional integral point of view", James Glimm, Arthur Jaffe or as derived here: ...
2
votes
1answer
837 views

Inverse square law in 2+1 dimensional universe from a Yukawa coupling?

There is a nice result that in 3+1 space time, a Yukawa coupling leads to an inverse square law force as the mass of the scalar field goes to zero. I was wondering what the corresponding force in a ...
6
votes
1answer
282 views

Noether's identities

I have some questions about the Noether's second theorem (generally not covered by field theory books): What is the most general Noether identity for (classical) field theories? Why are Noether ...
1
vote
0answers
67 views

Global part of a local symmetry?

What is exactly meant by "Global part of a Local symmetry"? What are its implications on a field theory at classical level? What are its implications at quantum level? How is it related to symmetry ...
1
vote
0answers
135 views

A fundamental equation for solitary wave and dimension analysis [closed]

According to the scalar Field theory we write Lagrangian as $$\mathcal{L}=\frac{1}{2}\partial^\mu \phi \partial_\mu \phi -\frac{m^2}{2}\phi^2 -\frac{\lambda}{4!}\phi^4 \tag {1}$$ What I want to do is ...
9
votes
1answer
282 views

Lagrangian for Goldstone mode + topological excitation

The XY-model Hamiltonian is the following, $${\cal H}~=~-J\sum_{\langle i,j\rangle} \cos (\theta_i -\theta_j).$$ The Goldstone mode corresponds to term $(\nabla \theta)^2$ in the effective ...
0
votes
0answers
106 views

Derrick’s theorem(2)

Related post : Derrick’s theorem Consider a theory in D spatial dimensions involving one or more scalar fields $\phi_a$, with a Lagrangian density of the form $$L= \frac{1}{2} G_{ab}(\phi) ...
0
votes
1answer
143 views

Vortex in D dimensions soliton

let us consider the two-dimensional configuration shown in Fig. 3.1a. The lengths of the arrows represent the magnitude of φ, while their directions indicate the orientation in the $φ_1 -φ_2$ plane. ...
0
votes
2answers
255 views

Derrick’s theorem

Consider a theory in D spatial dimensions involving one or more scalar fields $\phi_a$, with a Lagrangian density of the form $$L= \frac{1}{2} G_{ab}(\phi) \partial_\mu \phi_a \partial^\mu \phi_b- ...
1
vote
1answer
330 views

sine-Gordon equation

I have derived a solition equation (2 dimensions) from scalar field theory $$\varphi(x) = v\tanh\Bigl(\tfrac{1}{2}m(x - x_0)\Bigr),\tag{1}$$ and also I have got sine-Gordon equation for solition ...
0
votes
0answers
215 views

Domain wall and kink solutions from solitions equations

A general solition equation can be obtaion from scalar field theory $$\varphi(x) = v\tanh\Bigl(\tfrac{1}{2}m(x - x_0)\Bigr),\tag{92.6}$$ where $x_0$ is a constant of integration when we drived this ...
2
votes
3answers
540 views

What is a nonlinear field?

I have read two possible definitions. A nonlinear field is A field taking values on a manifold. A field whose equation is nonlinear. What do you understand by a nonlinear field or a nonlinear ...
-1
votes
1answer
328 views

Double- well potential and Mexican potential

Is double well potential related to Maxican hat potential? I have found on Quantum Field Theory in a Nutshell by A. Zee He wrote the double well potential as : $V (φ) = (λ/4)(φ^ 2 − v^2)^2$. Can ...