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58 views

How to “transfer” indices from dot product to metric?

In this source, the author (Andrzej Pokraka, Solutions to problems from Peskin & Schroeder) is computing an integral related to scalar QED. In the step where the equation is labelled (29), the ...
4
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1answer
57 views

Transformation of the derivative of the scalar field in Ramond's book about QFT

In the book by Pierre Ramond about quantum field theory, he explores in chapter 1.4 (p.13) the behavior of fields under Poincaré transformations. He starts by explaining that infinitesimal ...
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0answers
40 views

Regarding notation used for infintesimal parameters of the Lorentz algebra and generators of the Lorentz group

I have a confusion regarding the notation that is used for infintesimal Lorentz transformations and the parameters that define the Lorentz transformation (used in various books such as Srednicki's and ...
1
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1answer
166 views

Gravitons and self-interaction

In the book quantum field theory and standard model by Schwartz, there is a problem 9.4 that says by considering lorentz invariance of Compton scattering, you can prove that for spin 1 massless field ...
4
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1answer
88 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 ...
0
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1answer
58 views

Why are fields which do not transform in a certain way not fundamental?

When I was first exposed to Math physics textbooks and textbooks on vector calculus, I found: Temperature distribution in a room $T(x,y,z,t)$ or the density variation in a fluid $\rho(x,y,z,t)$ etc ...
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1answer
88 views

Is this a right approach to show that $\partial_{\mu} \phi \partial^{\mu} \phi $ is Lorentz Invariant?

When trying to convince myself that $\partial_{\mu} \phi \partial^{\mu} \phi $ is Lorentz Invariant, I stumbled upon this approach: The last equation should read - $\partial_{i} \phi \partial^{i} \...
2
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2answers
147 views

Does the electromagnetic field have a “rest mass” that is conserved?

In an answer to this Physics SE question, @ChiralAnomaly demonstrated that, indeed, there is a minimum field energy density observable at any point in an EM field. With a bit more calculation, it's ...
5
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2answers
314 views

Poincare transformations and “three kinds of infinitesimal variations”

I'm currently reading these$^1$ lec. notes as an introduction to relativistic QFT. In chapter two (pp.57-61) he introduces the concept of field variations along with some formulas for the different ...
0
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2answers
129 views

Finding the expression for probability density (the Klein Gordon equation)

Source: Quantum Field Theory for the Gifted Amateur by Tom Lancaster, Stephen J. Blundell. I am struggling to understand the logical step from the outline of the 'proof' in the footnote, to the fact ...
1
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2answers
88 views

What does $U^{-1}(\Lambda)\phi(\Lambda y) U(\Lambda) = \phi(y)$ physically mean?

In QFT, let $U(\Lambda)$ denote a unitary representation of the Lorentz group. Let $\phi(x^{\mu})$ be scalar field operators in the Hilbert space; in other words: $$U^{-1}(\Lambda)\phi(x^{\mu}) U(\...
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0answers
55 views

Retrieving the non-relativistic formulas for electric and magnetic fields from relativistic formulas

I was checking the formulas for the electric and magnetic fields components E and B given in this link from Wikipedia: https://en.wikipedia.org/wiki/Classical_electromagnetism_and_special_relativity#...
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0answers
37 views

A Scalar Function Tranformation — Question on Notation in 't Hooft Document

I started reading a document by Gerard 't Hooft which can be found here. Right at the start I am puzzled by a simple expression. It is equation 2.2 showing how a scalar function transforms. I ...
2
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2answers
138 views

$\partial^{\nu} \partial_{\nu}$ vs. $\partial_{\nu} \partial^{\nu}$

I was doing a problem regarding field theory. I am given the following lagrangian density: $$\mathcal{L}=\frac{1}{2}\partial_\mu\phi_i\partial^\mu\phi_i-\frac{m^2}{2}\phi_i\phi_i$$ for three scalar ...
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0answers
56 views

Variation of vector field under Lorentz transformation and gauge transformation

In a paper I am reading, it is stated that under a Lorentz transformation, the coordinates transform as $x^{\mu} \to \Lambda^{\mu}_{\nu}x^{\nu}$, and so the change in the (vector) field at the same ...
2
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2answers
133 views

How is $\int \frac{d^{3}\mathbf{p}}{(2\pi)^3}\frac{1}{2\sqrt{|\mathbf{p}|^2+m^2}}$ manifestly Lorentz-Invariant?

When writing integrals that look like $$ \int \frac{d^{3}\mathbf{p}}{(2\pi)^3}\frac{1}{2\sqrt{|\mathbf{p}|^2+m^2}} \ = \int \frac{d^4p}{(2\pi)^4}\ 2\pi\ \delta(p^2+m^2)\Theta(p^0) $$ it is often said ...
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2answers
55 views

Conservation of improved energy momentum tensor of a real massless scalar field

So I'm supposed to find that the improved energy momentum tensor of the scalar field $\phi$ satisfying the evolution equation $\Box \phi = 0$ is conserved. The improved energy momentum tensor is: $T^{...
1
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4answers
217 views

What causes $A^{\mu\nu}_{\pm}=F^{\mu\nu}\pm i \tilde{F}^{\mu\nu}$ to have three independent components rather than six?

Both the elctromagnetic field strength tensor $F^{\mu\nu}$ and its dual $\tilde{F}^{\mu\nu}$ defined as $\tilde{F}^{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\lambda\rho}F_{\lambda\rho}$ are examples of ...
2
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3answers
175 views

Lagrangian of EM field: Why the $B$-field term has a minus sign in front of it in the Lagrangian?

I know that $L = T - U$ and that, in the non-relativistic case $$L= \frac{1}2mv^2 - q\phi(r,t) + q\vec{v}\cdot\vec{A}(r,t).\tag{1} $$ My lecturer used the following form of the Lagrangian density ...
1
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2answers
44 views

Different weights for time and spatial derivative in Lagrangian Density

I'm new to QFT and trying to understand the form of the Lagrangian densitys used. As a simple model you often see a Lagrangian density of the form $${\mathcal L} = \frac{1}{2} \partial_j \phi_n \...
0
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1answer
125 views

Why is the Lagrangian $-\partial^u\overline\Psi\partial_u\Psi-m^2\overline\Psi\Psi$ not Lorentz invariant?

Let $-\partial^u\overline\Psi\partial_u\Psi-m^2\overline\Psi\Psi$ be a Lagrangian density. Here $\Psi$ is the Dirac spinor and $\overline \Psi$ is defined to be $\Psi^\dagger \gamma^0$. It is said ...
2
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1answer
44 views

From “Matrix” form to “Component” (tensor) form

Given $\omega=-\eta\omega^T\eta^{-1}=-\eta\omega^T\eta$, where $\eta$ is the usual Minkowski metric. Is the following logic correct?: $$ {\omega^{~\mu}}_{\nu}= -{\eta_{\varepsilon\nu}}{\left(...
-1
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1answer
65 views

Is there a name for a function or field of position and rotation?

A scalar field is a function which has a different value at all positions e.g. $\phi(x)$ where $x$ is a 3-vector. Imagine that the value of a field depended not only on the position of a detector but ...
3
votes
1answer
100 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 ...
1
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1answer
144 views

Weyl Spinors and Lorentz Invariance

Let $\phi_a$ and $\chi_{\dot{a}}$ be two component commuting spinors, where $\chi$ is an anti-spinor. In terms of some spinor basis, these can both be written in some arbitrary frame as $$ \phi_a(P) =...
1
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0answers
206 views

Explicitly proving that the Hamiltonian is Lorentz covariant

I want to show explicitly that the Hamiltonian $$ H = -\Omega V + \int d\tilde{\textbf{k}} \omega (a^\dagger(\textbf{k}) a(\textbf{k}) + b(\textbf{k}) b^\dagger(\textbf{k}) ) $$ is Lorentz covariant....
3
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0answers
269 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 ...
0
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0answers
59 views

Hilbert space of free one particle states; different formalisms?

I am having trouble understanding the formal setting of the one particle states as constructed in e.g. Weinberg QFT vol 1. Is the relevant Hilbert space $\mathscr{H}=L^2(\mathbb{R}^3)$? Weinberg ...
2
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1answer
91 views

How come $\frac{\partial(\partial_{\beta}A_{\gamma})}{\partial(\partial_{\mu}A_{\nu})} = g_{\beta\mu}g_{\gamma\nu}$?

For context, this equation is used in the following (from Schwartz's QFT 3.44) $$\partial_{\mu} \frac{\partial(\partial_{\beta}A_{\gamma})^2}{\partial(\partial_{\mu}A_{\nu})} = \partial_{\mu}\left[2(\...
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0answers
279 views

Helicity of Massless Particles

A well-known result of Wigner's classification of relativistic particles is that massless particles transform with helicity $h \oplus -h$ under $ISO(2)$. Thus, such particles have two helicity states. ...
-2
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1answer
133 views

Linear momentum of a Weyl spinor

Let's say we have a left-handed Weyl spinor as follows: \begin{equation} \chi = \begin{pmatrix} \alpha \\ \beta \end{pmatrix} \end{equation} where $\alpha$ and $\beta$ are complex components. What ...
5
votes
1answer
299 views

Why don't we consider the “most general” spin 1 Lagrangian, but only a special case?

The most general Lorentz invariant, renormalizable Lagrangian for a spin 1 field $A_\mu$ reads \begin{equation}\mathscr{L}_{\text{Proca}}= C_1 \partial^\mu A^\nu \partial_\mu A_\nu + C_2 \partial^\...
2
votes
2answers
133 views

Quantisation of Lorentz charge

A conserved charge can be derived from the Noether current corresponding to the Lorentz symmetry: $$ Q_i = \int d^3x (x_iT_{00}-tT_{0i}) $$ where $T_{\mu\nu}$ is the usual stress-energy tensor. ...
2
votes
1answer
169 views

Massive spin-$s$ representations of the Poincare group on the space of spin tensor fields

Context The following is from the book "Ideas and methods in supersymmetry and supergravity" by I.L. Buchbinder and S.M Kuzenko, pg 56-60. It is about realizing the irreducible massive ...
9
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1answer
356 views

Meaning of Lorentz Generators

I'm trying to understand infinitesimal Lorentz transformations in quantum field theory. I've studied some Lie theory from mathematicians, but I'm having trouble adjusting conceptually to how Lie ...
0
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0answers
55 views

How to find Bilinears of a theory?

I'm trying to understand how one finds the bilinears of a given theory. In most litterature the bilinears are not really derived but rather taken as fundamental. The dirac bilinears are of course: $$\...
0
votes
1answer
159 views

Non-relativistic and relativistic field theory Lagrangian confusion

In non-relativistic classical mechanics if I was to describe the waves within a three dimensional medium, I would write down a Lagrangian $L$ in terms of the Lagrangian density $\mathcal{L}$ as $$L ...
1
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2answers
97 views

Confusion about the mathematical nature of Elecromagnetic tensor end the E, B fields

I have quite a lot of confusion so the question may result not totally clear cause of that. I'll take any advice to improve it and I'll try to be as clear as possible. Everything from now on is what I ...
11
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2answers
760 views

Invariance under boosts but not rotations?

I am aware that there are 6 independent infinitesimal Lorentz transformations that can be separated into 3 rotations and 3 boosts. Is it possible for a quantum field theory to be invariant under the ...
-1
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1answer
38 views

How is the idea of the static electric potential $\phi$ extended relativistically?

In classical electrodynamics there's the electrostatic potential $\phi$ which can be differentiated wrt space to give the static electric field $\vec E$. Is this idea perhaps extended relativistically ...
0
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1answer
195 views

Local Lorentz Invariance and Conformal Metric Transformations

It is often repeated that Lorentz invariance of Special Relativity (i.e., allowed solutions to Maxwell's Equations) is proven experimentally. This is clearly the case for local experiments and ...
1
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2answers
239 views

Does it matter if Poincaré transformations are just coordinate changes?

In a previous question of mine it was established that special relativity may be formulated as the theory of a (smooth) Lorentz manifold $(M,g)$ that is diffeomorphic to (the standard diff. structure ...
2
votes
1answer
473 views

Conserved current of a Klein-Gordon field

This problem is closely related to Problem 7 in this problem set from David Tong's QFT course: http://www.damtp.cam.ac.uk/user/tong/qft/oh1.pdf So I am studying a Klein-Gordon field $\phi$ with ...
2
votes
1answer
329 views

Klein-Gordon field $\phi$ and a Lorentz transformation

I'm supposed to consider a Lorentz transformation of the form $$\Lambda^{\mu}_{\ \nu} = \delta^{\mu}_{\ \nu} + \omega^{\mu}_{\ \nu},$$ where $\omega$ is some tensor. Since Lorentz tranformations ...
1
vote
3answers
546 views

How precisely the Klein-Gordon equation is derived?

In various articles (and books) such as the wiki article of the Klein-Gordon equation wrote: "The Klein-Gordon equation is a "quantized" version of the relativistic energy-momentum relation"; In ...
6
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1answer
292 views

Why does one study the representations of Lorentz group instead of studying only the representations of Poincare group?

Why does one separately study the representations of the Lorentz group and the Poincaré group, instead of directly and only studying the representations of the Poincaré group? After all, the ...
0
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3answers
790 views

Do Photons Really “Exist” as Particles? [closed]

[Major Edit Begins] My question below (which I leave in for posterity) has fairly been flagged for clarification which I shall attempt while trying to include the information already received. If an ...
2
votes
1answer
134 views

Klein-Gordon inner product: how to make it real

While building its way up to the construction of an inner product, one stumbles upon the following equation: \begin{equation} \partial_i(\varphi_2^*(x)\overleftrightarrow{\partial^i}\varphi_1(x))=\...
0
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1answer
181 views

Covariant and contravariant derivatives in Klein-Gordon equation

Whilst exposing how a scalar product for the solutions of the Klein-Gordon equation (written as $(\Box + m^2)\varphi(x)=0$) can be derived, my textbook starts from the following system \begin{cases} \...
3
votes
3answers
372 views

Why under Lorentz transformations the Higgs boson is a scalar field and under $SU(2)$ it is a doublet?

I am a bit confused about this difference. My understanding is that when we build a $G$-bundle, where $G$ is a gauge group, we have a representation $\rho:G\to GL(V)$ that acts on the fibers of the $G$...