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26 views

Regarding Chebychev and Gravity

I recently learned Chebyshev's Inequality (in statistics). He uses the inverse square to specify an upper bound to the proportion of entities of a set as it recedes on either side of the mean. I am ...
2
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1answer
55 views

How to derive the conserved current of the Klein Gordon equation?

Similarly to the probability current in non-relativistic quantum mechanics, there is a conserved current for the Klein Gordon equation, however a different one. I'm trying to calculate that. The KG ...
0
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2answers
114 views

How can I prove that $\langle\Omega\vert \phi(x) \vert\Omega\rangle \langle\Omega\vert\phi(y)\vert\Omega\rangle=0$ for a scalar field?

From Peskin-Schroeder, p.212: The term $$ \langle \Omega | \phi(x) | \Omega \rangle \langle \Omega |\phi(y) | \Omega \rangle$$ is usually zero by symmetry; for higher-spin fields, it is zero ...
3
votes
1answer
97 views

Calculation of the Abelian Induced Chern-Simons Term

In Gerald Dunne's paper "Aspects of Chern-Simons Theory" (http://arxiv.org/abs/hep-th/9902115) I'm a little confused as to how equation (225) on page 53 is obtained. Equation (225): ...
4
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1answer
435 views

What is the fundamental representation in field theory?

In field theory we associate to each Gauge theory a continuous group of local transformations (a Gauge group), and then we require\define fermion fields to be irreducible representations belonging to ...
3
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0answers
84 views

Field Lagrangian <--> Particle Lagrangian

The action-functionals describing the motion $\mathbf{x}:[a,b]\to \mathbb{R}^3$ of a free particle of mass $m$ and the evolution $\varphi:[a,b]\times \Omega\to \mathbb{R}$ of a free scalar field of ...
1
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1answer
66 views

Obtaining momentum operator $P^\mu$ from Lagrangian and energy-momentum tensor $T^{\mu\nu}$

I am pretty new to quantum field theory. Given the Lagrangian density, $$ \mathcal{L} = \frac{1}{2} ( \partial_\mu \phi ) ( \partial^\mu \phi ) - \frac{1}{2} m^2 \phi^2 $$ and its energy-momentum ...
3
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1answer
105 views

Trying to show that the current is conserved

$ \newcommand{\p}{\partial} $ I am trying to show that the current $J^{\mu} = (\gamma_{\nu}\partial^{\nu} \phi - m\phi)\gamma^{\mu}\psi$ is conserved for all fields that satisfy the Klein-Gordon and ...
2
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0answers
42 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 ...
-1
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1answer
62 views

How one can know the gauge field emerging from the local gauge invariance is actually the EM field? [closed]

How one can know the gauge field emerging from the local gauge invariance is actually the EM field? I understood in a simple scalar field whose Lagrangian is given by $ \mathcal{L} = ...
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9answers
3k views

Why are differential equations for fields in physics of order two?

What is the reason for the observation that across the board fields in physics are generally governed by second order (partial) differential equations? If someone on the street would flat out ask ...
6
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1answer
175 views

Cut-off Regularisation and Renormalisation in Scalar Field Theory, Deriving the Cutoff Independent Physical Mass

I'm having trouble reproducing Equation 42: \begin{equation}\tag{1} m^{2}_{\text{phys}}= m^{2}_{r} + m^{2}_{r} \tilde{\lambda} \text{log} \left( \dfrac{m^{2}_{r}}{\mu^{2}} \right) \end{equation} ...
1
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1answer
78 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 ...
1
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1answer
84 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 - ...
2
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1answer
186 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}, ...
1
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0answers
59 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 ...
0
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1answer
80 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 ...
2
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4answers
165 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 ...
0
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1answer
46 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
59 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
74 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
177 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
134 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?
1
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0answers
97 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 ...
1
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1answer
105 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
129 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
110 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 ...
1
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1answer
91 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
68 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: ...
2
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0answers
52 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 ...
0
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1answer
75 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
126 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
73 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 ...
1
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0answers
54 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
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1answer
260 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 ...
3
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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 ...
1
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1answer
150 views
1
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0answers
79 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} ...
10
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3answers
293 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 ...
7
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0answers
112 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
392 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 ...
3
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0answers
78 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
87 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
389 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
211 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 ...
0
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0answers
55 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
312 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
114 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
89 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
229 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... ...