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57 views

Translation invariance of point particles as a field theory

The case of point particles, relativistic or not, can be treated as a field theory in general, ie for the $(1+1)$-dimensional case this is the theory of a field theory on the vector bundle $$\pi : \...
1
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2answers
76 views

Does anyone have references on classical field theory that develops the differential form formalism?

I am familiar with the usual way of doing Classical Field Theory, but I am currently taking a course where the professor works with differential forms to teach the subject. I wonder if anyone knows ...
1
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0answers
43 views

Extended objects, bundle description and transformations

In the hope of trying to come up with a clear mental picture for what a transformation is in physics, I encounter some difficulties due to the variety of objects that appear in physics. While no ...
2
votes
1answer
135 views

The equation of motion for a scalar field in curved spacetime in terms of the covariant derivative

The equation of motion for a scalar field in curved spacetime $$\frac{\partial\mathcal{L}}{\partial\phi}=\frac{1}{\sqrt{-g}}\partial_{\mu}\left[\sqrt{-g}\frac{\partial\mathcal{L}}{\partial\left(\...
2
votes
2answers
122 views

Yang-Mills Bianchi identity in tensor notation vs form notation

I've seen the Yang-Mills Bianchi identity written as both $$0 = dF^a + f^{abc} A^b \wedge F^c$$ and, in tensor notation, as $$\epsilon^{\mu\nu\lambda\sigma}D_{\nu} F^a_{\lambda\sigma} = 0.$$ Here ...
1
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1answer
93 views

What are Connections in physics?

This question arises from a personal misunderstanding about a conversation with a friend of mine. He asked me a question about the "truly nature" of spinors, i.e., he asked a question to me about what ...
2
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4answers
141 views

Inconsistency between $d_A = d + A \wedge$ and $d_A = d(..) + [A,..]$?

I am confused by something basic stated in this https://physics.stackexchange.com/a/429947/42982 by @ACuriousMind and some fact I knew of. Here $d_A$ is covariant derivative. $d_A A=F$ --- @...
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0answers
36 views

Simplify Yang-Mills Equation of Motion in the 1-form gauge field $A$

We know the Yang-Mills theory Equation of Motion (eom) without source $$ * D * F = * (d (* F ) + [A, (* F )])= 0. $$ My question is that what are the most simple form we can boil down this ...
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1answer
56 views

Mistake or Rewriting of Yang-Mills in Nakahara

I am familiar with Yang-Mills equation of motion E.O.M. (without matter or source fields) in differential form. $$ D * F =0 $$ and Bianchi identity $$ D F=0 $$ where $F= dA + A \wedge A$ and $D=d + [...
4
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0answers
63 views

Why is it that the equation of a massless scalar field *must* be conformal invariant?

I'm reading a paper [1], p.111 where it is said that: However, the equation of scalar field with zero mass must be conformal invariant while equation $\square\varphi=0$ does not satisfy this ...
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0answers
54 views

Covariant derivative of a composite field and the chain rule

I have a gauge theory with some rather strange covariant derivatives and I am wondering how they act on a composite field like $\psi= \phi\psi'$. In my setup, the covariant derivative acting on a ...
2
votes
1answer
93 views

Covariant derivative in field theory

I'm reading Physics from Symmetry by Jakob Schwichtenberg and in Chapter 7 he introduces the covariant derivative when deriving the interaction Lagrangian density for the spin-half - spin-1 field: $$ ...
2
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1answer
115 views

Gauge-invariance of Lagrangians

I am rereading David Bleecker's Gauge Theory and Variational Principles, and I have realized I don't understand something. The offending part is in 3.3 (page 50-52), however I am reproducing the ...
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2answers
146 views

Hodge dual and QED

I was studying the paper Topological massive gauge theories in three dimensions by Deser, Jackiw and Templeton. In the paper, they use Hodge dual for some reason which I don't understand at all. So I ...
1
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1answer
108 views

Obtaining Brans-Dicke theory scalar (wave) equation

I have trouble with obtaining d'Alambert equation for scalar field in Brans-Dicke gravity (http://www.scholarpedia.org/article/Jordan-Brans-Dicke_Theory). B-D gravity langrangian density is given by: ...
5
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1answer
155 views

What exactly are the sections in gauge theories?

In trying to understand precisely how fiber bundle theory maps to physical models, I came across this quotation: We can think of the elements of the principal bundle as generalized frames for the ...
2
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1answer
82 views

Prove that background independence and diffeomorphism invariance of a spacetime theory are equivalent

A theory's equations can generally be derived from an action along with a principle of least action ($\delta S=0$). The action is given by: $$ S[f_1, f_2, ...]=\int_M \mathcal{L}(f_1, f_2, ..., g_1, ...
4
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0answers
273 views

How can I identify gauge transformations of fields with gauge transformations on a principal bundle?

I have some trouble with identifying what we do in physics regarding fields and bundle theory. I start from the following construction which I hope is ok: with a Lie group G and a smooth manifold M I ...
2
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1answer
60 views

Computing the derivatives of a Lagrangian on a Riemannian manifold

Consider a $Riemannian\space n-dimensional\space manifold$ with coordinates {$x^i$} $(i=1,...,n)$. Let the arc length parameter be $t$. So that $\frac{d}{dt}x^i(t)\equiv\dot{x}^i(t)$ is the usual ...
1
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1answer
90 views

How to raise and lower indices of field equations in curvilinear coordinates?

I am confused about a basic question. Consider the field equation of a gauge theory written in terms of curvilinear coordinates $$\frac{1}{\sqrt{-g}}D_\mu(\sqrt{-g}F^{\mu\nu})=j^\nu.$$ My questions ...
2
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1answer
91 views

What is the essential difference between the orbifold $S_1/Z_2$ and an interval say, $[0,1]$?

As the title asked, what is the essential difference between the orbifold $S_1/Z_2$ and an interval say, $[0,1]$? I mean $S_1$ can be [-1,1] with -1 and +1 identified. Now $S_1/Z_2$ then is just [0,1]...
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0answers
53 views

Linear term in NLSM action, expansion in terms of geodesic tangent vectors

The current question has arised as I started reading the the book by Ketov. We define the NLSM action as (for simplicity, I also assume $g_{ab}=g_{ba}$ and also the vanishing torsion): \begin{...
1
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0answers
71 views

Is parity symmetry broken by all actions?

Any orientation reversed transformation, which includes the parity and time reversal transformation, changes the volume form as $$ \sqrt{-g}\mathrm{d}^4x ~\to~ - \sqrt{-g}\mathrm{d}^4x, $$ so all ...
3
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1answer
169 views

Physical interpretation of differential forms with values in $E$ when $E$ is a vector bundle whose sections are fields

Disclaimer: I'm much more a mathematician then a physicist and this question is slightly mathematical however it made more sense to ask it in this site then any other. Let $M$ be a manifold of ...
1
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1answer
77 views

Ambiguity with definitions of vector potential

In one of my books (the great Baez & Munian's "Gauge fields, knots and gravity"), the vector potential is defined as a $End(E)$ valued 1-form, with $End(E)$ endomorphisms of the fiber $E$. So, ...
0
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1answer
84 views

Supersymmetric transformations require flat connections?

We read at 0912.4261[end of page 9 and beginning of page 10] that any susy field configuration of a theory must obey the conditions implied by setting the supersymmetric variations of fermions ...
2
votes
1answer
175 views

Green's function for Dirac operator on $S^4$

Let $S^4$ be a round sphere of radius 1 (with the standard Riemannian round metric), and let $D_\text{F} \equiv \gamma^\mu \nabla_\mu$ be the Dirac operator on $S^4$, acting on the usual spinors for ...
0
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0answers
100 views

Calculating connections for fiber bundles,

I think I understand the concept of covariant derivative, connection, even what a fiber bundle is, but I have a very specific question: how do you calculate the connection for an specific group? I ...
3
votes
1answer
149 views

Nature of the “charge” coupling constant in QED

I am trying to understand classical gauge theory from a differential geometric point of view, but there is something I don't particularily understand. I think the most appropriate answer as to why we ...
5
votes
2answers
531 views

What is the differential-geometric formulation of field theories?

I understand that in classical mechanics, a system with $N$ degrees of freedom is modeled by an $N$ dimensional Manifold called the configuration space. We then look at the tangent and cotangent ...
3
votes
3answers
879 views

What is a nonlinear manifold?

The Wikipedia article defines a non-linear sigma model as model for a scalar field $\Sigma$ which takes on values in a nonlinear manifold called the target manifold $T$. What is the definition of a ...
8
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2answers
471 views

Any “connection” between uncountably infinitely many differentiable manifolds of dimension 4 and the spacetime having dimension four?

Self-studying general relativity, I came across a rather mind-blowing statement (for a beginner like me). Maybe this question is a naive one because of my lack of knowledge in differential geometry. ...
3
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1answer
277 views

Does a gauge group $G$ determine the Principal $G$-bundle?

I'm trying to understand the mathematical underpinnings of gauge theories in the language of principal $G$-bundles and associated vector bundles. Not long ago, I had assumed that the physical choice ...
1
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0answers
76 views

Master-level Minicourse on Topological Properties of $SU(2)$ Sphalerons

I am doing a master's thesis on the properties and applications of $SU(2)$ sphalerons. I am in desperate need of a source (textbook, lecture notes, etc.) through which I can learn the basics of the ...
9
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2answers
733 views

Yang-Mills potential and principal bundles

In section 2.7.2 of Bertlmann's "Anomalies in quantum field theory", it is stated that since a non-trivial principal bundle (based on a Lie group $G$) does not admit a global section, the Yang-Mills ...
2
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0answers
108 views

Construction of vector bundles of relativistic fields by Mackey's method of induced representation

I recently stumbled on Sternberg's book on group theory and physics. The ideas expressed in the book are really great, but the detailed reasoning is very hard to follow, I find. I am kind of stuck ...
2
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0answers
387 views

Jet bundles for physicists

In order to make Classical Field Theory rigorous we need the idea of jet bundles. I've seem some books on the subject, but most of them are aimed at mathematicians and tend to go quite deep in the ...
1
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1answer
288 views

ADHM construction and Momentum Map

while I was reading about ADHM construction I had some troubles with precise geometrical identification of the various quantities. My doubts is well manifest in these two Wikipedia pages 1) ADHM ...
8
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1answer
1k views

Use partial or covariant derivatives when deriving equations of a field theory?

I feel like this question has been asked before but I can't find it. would the Euler Lagrange equation for, say, the standard model Lagrangian be $$\frac{\partial L}{\partial \phi}=\partial_\mu \frac{\...
4
votes
1answer
175 views

Non-Euclidean mechanics; is it useful?

Special relativity has the following single-particle Lagrangian: $$S = \int_{t_0}^{t_f}\sqrt {\langle \mathrm d\vec{s},\mathrm d\vec{s}\rangle}.$$ Clearly it is based on Euclidean norms; it is in ...
11
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0answers
128 views

Is it known what the necessary and sufficient conditions are for the existence of a “3+1 split” (by means of a foliation) of a (Lorentzian) manifold?

When trying to do physics on a more general pseudo-Riemannian manifold we want to require that there is a foliation of this manifold into three-dimensional subspaces. By this I mean we would like to ...
36
votes
1answer
2k views

What, to a physicist, are instantons and the Donaldson invariants?

I study gauge theory from a mathematical perspective. To me, one of the most fundamental ideas is the notion of an instanton on a 4-manifold. To be precise, I have a Riemannian 4-manifold and a ...
4
votes
2answers
254 views

Field theory where fields are differential forms, other than electromagnetism [closed]

I am looking for a few examples of field theories (classical or quantum) that can be formulated taking the fields to be differential forms at least of degree 1 (not counting 0-forms) excluiding ...
1
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0answers
79 views

Is the coordinate transformation of an object the same of the action of a group on this same object?

I am having troubles in understanding frame transformations in physics from the mathematical point of view. What I understand for a coordinate transformation is just a function to one chart to another ...
1
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1answer
216 views

Classical Field Theory Using Geometry

I would like to know if there are good introductory courses on Classical Field Theory taught in a differential geometry approach yet one doesn't need a background in those mathematical subjects but ...
5
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2answers
303 views

General relativity without curvature?

Is there a reformulation of general relativity without curved space time, just with fields (like classical E&M)? Edit: removed the part about E&M with curvature (multiple posts).
0
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1answer
120 views

In field theory, why are some symmetry transformations applied to the field values while other act on the space that the fields are defined on?

My basic understanding is that a field theory consists of symmetry groups, a space $S$ that the symmetry groups act on and of fields defined on that space $S$. In other words, the space $S$ is the ...
3
votes
1answer
304 views

Is EM interpreted in a principal or vector bundle?

I've read in a few places that EM is a $U(1)$-principal bundle; but is this correct? Isn't it rather an associated vector bundle using the adjoint representation of $U(1)$?
4
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1answer
956 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 \left(\frac {\partial L}{\partial\left(\partial_\...
16
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3answers
2k views

Global vs. local gauge group in mathematical sense - physics examples?

Upon reading about the principal bundle picture of (quantum) field theory I encountered two different definitions of the gauge group: Local gauge group $G$. Corresponds to the fibers of the $G$-...