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1answer
64 views

Question about global internal $SO(n)$ symmetry

I have the following Lagrangian (density) for bosons $$L = \partial_{\mu} \phi^i \partial^{\mu}\phi^i+ m^2\phi^i \phi^i$$ and I am trying to understand why this Lagrangian is invariant under ...
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1answer
52 views

Understaning Euclidean Green's function

Consider a scalar field coupled to a source $$(\Box - m^2)\phi(x) = -J(x)\tag{1}.$$ Then, the response of the source is determined by the Green's function $G(x-y)$, which satisfies $$(\Box - ...
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1answer
157 views

Spin tensor and Lorentz group operator in bispinor case

For infinisesimal bispinor transformations we have $$ \delta \Psi = \frac{1}{2}\omega^{\mu \nu}\eta_{\mu \nu}\Psi , \quad \delta \bar {\Psi} = -\frac{1}{2}\omega^{\mu \nu}\bar {\Psi}\eta_{\mu \nu}, ...
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0answers
51 views

Difference between Gravitational and Matter Scalar Fields

In the context of Scalar-Tensor theories of gravity (for example in Brans-Dicke) what is the difference between gravitational and matter scalar Fields? My doubt comes from "The scalar-tensor Theory ...
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1answer
65 views

Is there a field for which neutral particle and antiparticle, can be considered as positive and negative charge?

I apologize, but QFT is not my domain. What I ask is connected with the question Do the fields exist without charges? . By analogy with the electron and proton, that carry the electric charges of the ...
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4answers
126 views

Space orientation of light waves

Recently I've started to be really intrigued with the electromagnetic spectrum and bumped into this problem: According to the wave theory of light (or any electromagnetic wave, really), the magnetic ...
1
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1answer
30 views

Why Do Stark Manifold Graphs All Have Negative Energy?

I have been studying Rydberg-Stark State Atoms and their Stark Manifolds (like the one on Wikipedia: http://en.wikipedia.org/wiki/File:Hfspec1.jpg) and I was wondering, Why does the y-axis (of Energy ...
2
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1answer
52 views

Fast and slow modes, and the vanishing of certain diagrams during re-normalization

In the middle of pg. 452 of Atland and Simonss Condensed Matter Field Theory, they state the following: Terms of $\mathcal{O}(\phi _{\text{s}}^3\phi _{\text{f}})$ do not arise because the addition ...
3
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1answer
67 views

Maxwell's Inspiration to think about fields

I was looking at a Wikipedia article which had the following statement Atomists, notably James Clerk Maxwell and Ludwig Boltzmann, applied [...]. In modern literature Maxwell is often thought ...
3
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1answer
112 views

Relationship between the on-shell and BPHZ renormalization schemes

In his book Quantum Field Theory - A Tourist Guide for Mathematicians, Gerald Folland introduces the on-shell renormalization scheme for the $ \phi^{4} $-scalar field theory. According to my ...
6
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1answer
125 views

Gauge fixing of an arbitrary field

How to count the number of degrees of freedom of an arbitrary field (vector or tensor)? In other words, what is the mathematical procedure of gauge fixing?
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0answers
60 views

Relation between $f(R)$ gravity and Tensor–vector–scalar (TeVeS) gravity

We know that there is a relation between f(R) gravity and scalar-tensor gravity. By applying the Legendre-Weyl transform, we can receive brans-dicke gravity from $f(R)$ gravity. If we start with the ...
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1answer
80 views

Scalar field in a Schwarzschild metric

I have found this article recently published in Classical and Quantum Gravity giving the exact solution of a scalar field in the Kerr-Newman metric. These authors also derived Hawking radiation for ...
4
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1answer
91 views

Total derivative in action of the field theory

Consider a classical field theory. When applying the least action I see that a term is considered total derivative. We say that $$\int \partial_\mu (\frac {\partial L}{\partial(\partial_\mu ...
3
votes
2answers
94 views

Scalar operators In Quantum Field Theory

I am trying to learn Quantum Field Theory and I am stuck in a basic point. What is the definition of a scalar operator in QFT? That is, how does it transform under a Poincare transformation? Why do ...
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1answer
77 views

Creation and annihilation operators

In our lecture today, we introduced two kinds of creation and annihilation operators. I want to restrict myself to the antisymmetric case: The first operator $a_k^{\dagger}$ creates a state ...
2
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1answer
50 views

Introduction of the vector potential $A_{\mu}$ for the local gauge invariance of the complex scalar field lagrangian [duplicate]

In Ryder, when trying to restore the local $U(1)$ gauge symmetry of the complex scalar field $\phi=\phi_1+i\phi_2$, the final Lagrangian consists of the following four parts: ...
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4answers
2k views

Lagrangian to Hamiltonian in Quantum Field Theory

While deriving Hamiltonian from Lagrangian density, we use the formula $$\mathcal{H} ~=~ \pi \dot{\phi} - \mathcal{L}.$$ But since we are considering space and time as parameters, why the formula ...
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0answers
49 views

Shouldn't we use a Hamiltonian that doesn't give special treatment to time? [duplicate]

If we have a Lagrangian $\mathcal L$ that depends on some scalar field $\phi$, we define the momentum as $\pi \doteqdot {\partial \mathcal L \over \partial \dot \phi}$. The Hamiltonian then is ...
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1answer
52 views

Complex scalar field

In his book on Quantum Field Theory, Ryder mentioned in p. 91 under the title Complex Scalar Fields and Electromagnetism, the following: He said that under a global phase transformation $$\phi ...
3
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1answer
98 views

Rigorous version of field Lagrangian

In Classical Mechanics the configuration of a system can be characterized by some point $s\in \mathbb{R}^n$ for some $n$. In particular, if it's a system of $k$ particles then $n = 3k$ and if there ...
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0answers
50 views

Difference between a “source dipole” and a “force dipole”

I know quite well what a dipole is and in general what multipole moments are (in the context of, for instance, electrodynamics). What I find myself confused by is something called a "force dipole" in ...
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0answers
56 views

In SUSY, why do fermions and gauge bosons in the same multiplet both transform in the adjoint representation of the gauge group?

I'm trying to understand a certain point about supersymmetry. We are dealing with a N=1 (i.e, one supersymmetric flavour), massless, four dimensional theory. Then the vector multiplet consists of a ...
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0answers
47 views

Mixed two-point vertex in QFT

I am considering a theory with two fields, say $\phi$ and $\psi$. The Lagrangian contains quadratic terms, i.e., propagators for both fields and a quartic interaction term for one of the fields. ...
3
votes
1answer
229 views

Which transformations *aren't* symmetries of a Lagrangian?

As far as I understand, Noether's theorem for fields works, as explained in David Tong's QFT lecture notes (page 14) for example, by saying that a transformation $\phi(x) \mapsto \phi(x) + \delta \phi ...
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2answers
2k views

Deriving Lagrangian density for electromagnetic field

In considering the (special) relativistic EM field, I understand that assuming a Lagrangian density of the form $$\mathcal{L} =-\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + \frac{1}{c}j_\mu A^\mu$$ and ...
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1answer
112 views
1
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0answers
65 views

What's the conserved stress energy tensor? [closed]

I've worked on this problem for forever and still don't really see the solution. Any help appreciated. Say we have the Lagrangian for a scalar field that's $U(1)$ charged,$$\mathcal{L} ...
9
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3answers
258 views

Why is fundamental physics taught in terms of particles?

According to this paper, there can be no relativistic quantum theory of localizeable particles ("relativity plus quantum mechanics exclusively requires a field ontology"). Sean Caroll has also argued ...
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0answers
96 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 ...
1
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2answers
335 views

Can a scalar field model gravity? How accurate would be the results? Are there any difficulties with such a model?

Newtonian gravity can be described by the equation: $$ \nabla^2 \phi = 4 \pi \rho G $$ where $\rho$ is the mass density, $\phi$ is the gravitational potential, and G is the universal gravitational ...
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2answers
119 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 ...
3
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0answers
68 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 ...
1
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1answer
82 views

Interpretation of $\vec{x}$ in QFT

I am still at an early stage of studying Quantum Field Theory (I am reading QFT In A Nutshell by A. Zee). In the book I'm reading, it starts from a discrete lattice of material "lumps" labeled by ...
2
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4answers
263 views

Poincare invariant Lagrangians

The Lagrangian density of a Poincare invariant theory should not depend explicitly on the space-time coordinates. Does this mean $$ \partial_\mu \mathcal{L}=0~? $$ If this is the case doesn't the ...
2
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0answers
148 views

Show: Lorentz-invariance of solution of Klein-Gordon equation [closed]

Assume $\psi$ is a solution of the Klein-Gordon equation (KGE). Let $\Lambda$ be a Lorentz transformation. Show: $\phi = \psi(\Lambda^{-1} \cdot )$ is also a solution of the KGE. I try to ...
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0answers
53 views

Definition of force in a scalar field theory

How do we define the force for a general scalar field theory? In particular what is the scalar force of the below equation of motion: where $\tilde{T}$ is the energy stress tensor of matter and $A$ ...
6
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1answer
219 views

Mathematical interpretation of Poisson Brackets

Lets say we are working in a classical scalar field theory and we have two functional $ F[\phi, \pi](x)$ and $G[\phi, \pi](x)$. In most of the references, starting with two functional the Poisson ...
0
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1answer
102 views

Conserved current for a constant translation of a free massless scalar field

In Zinn-Justin's Quantum Field Theory and Critical Phenomena they start with an action for a free massless scalar field: $$S(\varphi) = \frac{1}{2}\int ...
2
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1answer
74 views

A Variation on Laplace's equation (context: Yang-Mills N-Instantons, Rajaraman's book)

Statement of the problem I need to solve the equation \begin{align} 0 = \frac{1}{\phi} \partial_{\sigma}\partial_{\sigma} \phi \hspace{20mm} (1) \end{align} where $\phi$ is a scalar field and ...
5
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2answers
170 views

Energy and momentum as partial derivatives of on-shell action in field theory

According to L&L, if we fix the initial position of a particle at a given time and consider the on-shell action as a function of the final coordinates and time, $S(q_1, \ldots, q_n, t)$, then... ...
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0answers
40 views

Total Vs Partial in Lagrange density?

I have a question regarding the red term below. This is the integration by parts during the derivation of the Euler-Lagrange equation for continuous systems. Why is this not the time derivative ...
3
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2answers
201 views

Classical Field Theory - Continuum limit in forming the Lagrangian density and the elasticity modulus

I have been looking at taking the continuum limit for a linear elastic rod of length $l$ modeled by a series of masses each of mass $m$ connected via massless springs of spring constant $k$. The ...
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1answer
47 views

Mass of small fluctuation around vacuum

For a potential $V$, how do we define the mass of a small fluctuation around its vacuum? For example I have the potential $$ V_\mathrm{eff}(\phi) = \frac{1}{2} \left(\frac{\rho}{M^2} - \mu^2\right) ...
1
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1answer
81 views

Hermiticity of the quantum field

The quantum field resultant from the quantization of a real classical field is hermitian, but why the quantum field corresponding to a complex classical field should be non-hermitian?
2
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0answers
115 views

N=4 SYM from Klebanov-Witten field theory

This is with reference to M. J. Strassler's lectures on "The Duality Cascade" pg. 46. I want to see how $\mathcal{N}=4$ SYM emerges when D3 branes, in the KW setup, are placed at smooth point of the ...
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2answers
196 views

Two expressions for potential energy in the gravitational field of the earth

Let $M$ be the mass of the earth, considered as a point mass, then the potential energy of a point with distance $r$ away from the center (assume $r > \textrm{radius of earth})$ is $$ U(r) = ...
4
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1answer
169 views

Spontaneous Symmetry Breaking

In Spontaneous symmetry breaking we have got that, a field $$\phi= \pm \sqrt{\frac{-m^2}{\lambda}}.$$ Now in order to get the unstable minima we need to guess the mass $m^2 <0 $. But can mass be ...
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0answers
47 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 ...
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2answers
314 views

(Un)countability in QFT

I am a mathematician self-studying physics, and a currently working on QFT with Srednicki's book. One thing that bothers me is that for a scalar field (in the Hamiltonian version) there is a ...