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4
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2answers
925 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? ...
1
vote
1answer
990 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 ...
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
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
720 views

Scalar field lagrangian in curved spacetime

I am studying inflation theory for a scalar field $\phi$ in curved spacetime. I want to obtain Euler-Lagrange equations for the action: $$ I\left[\phi\right] = \int ...
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
97 views

Definition: Coupling [closed]

What does it mean to say that 2 fields are coupled? More generally, what does "coupling" mean?
2
votes
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 ...
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
0answers
496 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 ...
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
3answers
335 views

Dimensions in lagrangian potential

According to Mankowski flat space dimensions We can write, $$L= \int \text{dt} \text d^d{x} \left[ \frac{1}{2} \dot\phi^2 - \frac{1}{2} \left(\frac{\partial \phi}{\partial r} \right)^2 ...
1
vote
1answer
151 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 ...
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 ...
1
vote
0answers
68 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 ...
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
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- ...
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. ...
1
vote
1answer
332 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
216 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 ...
-1
votes
1answer
145 views

Symmetry breaking with Lagrangian

I have been studying the spontaneous symmetry braking from Zee (Quantum Field theory ) and found in the page 224, he wrote the lagrangian as $$\mathcal{L}= \frac{1}{2}\{ λ (∂φ)^2 + μ^2φ^ 2\} − ...
-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 ...
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 ...
6
votes
1answer
283 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
2answers
124 views

Does spatial coupling prohibit resonances due to an external source field?

The harmonic oscillator coupled to a sinodial external source $$\frac{\partial^2 x(t)}{\partial t^2}+\omega_0^2 x(t)=F_0\sin(\omega_\text{ext}\ t),$$ has the solution $$x(t)=x(0)\cos(\omega_0 t)+C ...
1
vote
1answer
136 views

Comparing interaction potential in standard $ϕ^4 $theory

I am posting this question again because, Willie Wong asked me to do it. So it is a continuing post of the Interaction potential in standard ϕ4 theory. I have been studying about solitions so I had ...
-2
votes
1answer
801 views

$\phi ^4$ theory explaining [closed]

In $φ^4$ theory we often write the Lagrangian as $$\mathcal{L}=\frac{1}{2}\partial^\mu \phi \partial_\mu \phi -\frac{m^2}{2}\phi^2 -\frac{\lambda}{4!}\phi^4 \tag {1}$$ If I want to write from the ...
1
vote
1answer
194 views

Potential in Relativistic Scalar Field Theory

My intention is to establish a Soliton equation. I have cropped a page from Mark Srednicki page no 576. I have understand the equation (92.1) but don't understand that how they guessed the ...
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
429 views

From Lagrangian to Hamiltonian in Fermionic Model

While going from a given Lagrangian to Hamiltonian for a fermionic field, we use the following formula. $$ H = \Sigma_{i} \pi_i \dot{\phi_i} - L$$ where $\pi_i = \dfrac{\partial L}{\partial ...
5
votes
0answers
75 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 ...
0
votes
1answer
281 views

Why do fields decrease with distance? [duplicate]

For example, electric, gravitational field decreases with $1/r^2$. Is it like decrease of energy of an object when goes it is moving with friction/air drag etc? Does it mean that field's strength is ...
8
votes
1answer
2k 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 ...
0
votes
1answer
92 views

How the nonlinear equation can be written like this?

We consider a scalar theory in a $1+D$ dimensional flat Minkowski space-time, with a general self-interaction potential, whose action can be written as \begin{equation} A=\int dt\, d^D\! x ...
1
vote
1answer
229 views

Interaction potential analysis from $\phi^4$ model

In this paper, the authors consider a real scalar field theory in $d$-dimensional flat Minkowski space-time, with the action given by $$S=\int d^d\! x ...
0
votes
1answer
125 views

Oscillon and soliton

I want to know the major difference between oscillon and soliton in terms of radiating energy with respect to time and position. And what about their localization?
3
votes
1answer
190 views

Definition of Local Function

Now a days I am studying Srednicki's QFT book. In its third chapter it is written that Any local function of φ(x) is a Lorentz scalar, [...] . Now my question is: What is a local function?
-1
votes
1answer
192 views

Linear/ non linear Scalar field theory

How do I understand that the action for the free relativistic scalar field theory is non linear? What will be the associated interaction potential of that equation?
-1
votes
1answer
331 views

Creation and Annihilation operator [closed]

In this page I want to know, why the equation (1.32) introduced creation and annihilation operator. Please elaborate.
0
votes
2answers
290 views

Difficulties with bra and ket notation

I have problem in understanding equation (1.23), I croped this image from Mark_Srednicki "Quantum field theory". Can anyone show me the reason for the equation (1.23)?
0
votes
1answer
228 views

Scalar field lagrangian and potential

This question is a continuation of this Phys.SE post. Scalar field theory does not have gauge symmetry, and in particular, $\phi\to\phi−1$ is not a gauge transformation. but why? and I want see the ...
2
votes
0answers
52 views

Is Inflation modelled by a field?

If Inflation is modelled by a field - is this a classical field or a quantum field? If classical are there good reasons not to quantise it? What are the implications of such a quantisation?
2
votes
2answers
242 views

Does a constant factor matter in the definition of the Noether current?

This is a very basic Lagrangian Field Theory question, it is about a definition convention. It takes much more time to typeset it than answering, but here it is: Consider a field Lagrangian with only ...
1
vote
2answers
191 views

In Noether's theorem, what is a “classical solution of the equations of motion”?

I'm reading a book which states that: for each generator of a global symmetry transformation, there is a current $j^{\mu}_{a}$ which, when evaluated on a classical solution of the equations of ...
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 ...
8
votes
3answers
769 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 ...
1
vote
0answers
92 views

relevant 4-dimensional theory with interacting vector field

A simple langragian that gives the simplest interaction is $\mathcal{L}=(\partial\phi)^2+(m\phi)^2$ where $m$ is some constant. Does anyone know of theory in four dimensions which is physically ...
2
votes
2answers
1k views

Pair production - mathematically?

Allover the web i am only seeing a statement similar to this: Pair production is not possible in vaccum, 3rd particle is needed so that conservation of momentum holds. Well noone out of many ...
10
votes
1answer
418 views

Auxiliary fields in supersymmetry

I know that auxiliary fields can be used to close the supersymmetry algebra in case the bosonic and fermionic on-shell degrees of freedom do not match. Could somebody please elaborate on this concept ...