We say that something is symmetric if there is some transformation we can perform on that object that leaves some property unchanged. The set of symmetry transformations of an object form a group, and the name of this group is used as the name of the symmetry of the object.

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Lagrangian is isotropic in space

In Landau & Lifshitz Mechanics, while deriving the properties of Lagrangian of a free particle in inertial frame, he uses the following points $:$ As space is homogeneous in inertial frame, a ...
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Symmetries, source terms, boundary conditions

If I recall correct you can say that e.g. the electric vectorfield is only a function of the radius if the source terms (charge) is spherical and uniform so that a group action that rotates space ...
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134 views

Noether's Theorem for Hamiltonians and Lagrangians

Looking around I see one version of Noether's Theorem that creates conserved quantities from symmetries that preserve the Lagrangian (e.g. http://math.ucr.edu/home/baez/noether.html), and another ...
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Why are there only two linearly independet quartic Higgs terms for the adjoint $24$ in $SU(5)$ GUTs?

I've read the statement in countless papers, for example, here Eq. 4.2 or here Eq. 2.1 without any further explanation or reference, that the "most general renormalizable Higgs potential" for an ...
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58 views

Strong interaction under $SO(3)$ isospin transformation

I'm given the following strong interaction: $$S = \int d^{4}x [\frac{1}{2} \partial_{\mu} \phi^{a} \partial^{\mu} \phi^{a} - \frac{m^2}{2} \phi^{a} \phi^{a}] ,\qquad a = 1,2,3 \text{.}$$ It is stated ...
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35 views

Gauss law question with regard to this example

I am really confused in Gauss law. Why do E3 and E2 pointing up? and also E1 pointing down? The lecture note said infer from symmetry and you will get the following but I dont really understand. ...
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Translational invariance implying diagonal representation in momentum space

I have just come across something in my reading of Peskin and Schroeder that claims that because a function, in this particular case a two-point correlation function, is translationally invariant, it ...
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36 views

The elementary particles uniformity and its limits in the context of matter [closed]

We know that matter particles are uniform, i.e. they are absolutely identical (1, 2, 3). Particles of various properties are uniform. But if we look at bigger matter elements, when and how does the ...
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98 views

Help on understanding a concept in Noether's first theorem

Given a Lie group $G$, whose most general transform depends on $\rho$ parameters, under the action of which an integral $I$ is invariant, there are $\rho$ linearly independent combinations of the ...
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Discrete translational invariance of lattice systems and conserved quantities [duplicate]

Imagine a crystal lattice with discrete translational symmetry. Is there any way to obtain local periodic conserved quantities by taking a derivative (deliberately left abstract)? The discretised ...
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72 views

Lagrangians not related via a total time derivative lead to same Noether symmetries?

Having answered my initial two questions (v1), I now consider a third possibility. Consider two Lagrangians that both lead to equivalent equations of motion. Suppose that they are not related via a ...
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112 views

Spin 1/2 wavefunction transformation under inversion and mirror symmetry

I'm considering group-theory applications to condensed matter physics now. In particular I work with the following paper: http://journals.aps.org/pr/pdf/10.1103/PhysRev.100.580 and try to understand ...
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102 views

Galilean invariance/ scale invariance of KPZ

I have problems with understanding what the Galilean invariance of KPZ means and how it is connencted to KPZ scale invariance? How can I see that KPZ is scale invariant? Why this symmetry impose ...
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34 views

C, P and T for Klein-Gordon Field

Taking transformation of Klein-gordon field under C, P and T as $$\phi_{p}(t,r)= \exp(i \alpha_{p}) \phi (t,-r)\ ,$$ $$\phi_{c}(t,r)= \exp(i \alpha_{c}) \phi^\dagger (t,r)\ ,$$ $$ \phi_{T}(t,r)= \...
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1answer
78 views

Work out components $F^{01}$ and $F^{ij}$ of the antisymmetric tensor $F^{\mu\nu}$ under the Lorentz Transform [closed]

Work out explicitly how the components $F^{0i}$ and $F^{ij}$ of the antysymmetric tensor $F^{\mu\nu}$ introduced in chapter I.6 transform under a Lorentz transformation This problem is from Zee, ...
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688 views

How to prove a symmetric tensor is indeed a tensor?

Our professor defined a rank $(k,l)$ tensor as something that transforms like a tensor as follows: $$T^{\mu_1' \mu_2'...\mu_k'}{}_{\nu_1'\nu_2'...\nu_l'} ~=~ \Lambda^{\mu_1'}{}_{\mu_1}...\Lambda^{\...
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79 views

Bondi-Metzner-Sachs (BMS) related Question(s)

I started studying the BMS group in connection with the set of papers by A. Strominger et al., also related with the supposed solution of the "Black Hole Information Paradox" by S. W. Hawking http://...
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339 views

Many Body Physics: Hamiltonian block structure and Symmetries

Consider a many body problem of a small cluster, e.g. the 'Hubbard-Cluster' (albeit the question may be of relevance for other Hamiltonians as well): $$\mathcal{H}=\sum_{<ij>\sigma} t_{ij} (c^\...
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385 views

When I take a Gaussian surface inside an insulating solid sphere, why does the outer volume have no effect on the electric field?

Say I try to find the magnitude of the electric field at any point within an insulating solid sphere. I know that in the case of a conductor, the electric field within it is 0. However, I have not ...
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83 views

Is there a systematic way to obtain all conserved quantities of a system?

I'd like to know whether, given a system, there's a way to obtain all the conserved quantities. For instance if the system consists of electric and magnetic fields, the fields must satisfy Maxwell's ...
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526 views

Why exactly do sometimes universal covers, and sometimes central extensions feature in the application of a symmetry group to quantum physics?

There seem to be two different things one must consider when representing a symmetry group in quantum mechanics: The universal cover: For instance, when representing the rotation group $\mathrm{SO}(...
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196 views

Difference between symmetry and invariance

I'm wondering what's the real difference between symmetry and invariance in Physics? I believe that sometimes the two words are given the same meaning and some other times they are used in a different ...
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124 views

Conservation Laws and Symmetry

The toughest of topics in physics, like Quantum Mechanics, Relativity, String theory, can be explained in layman words and many have done so. Though there is no substitute to the understanding a ...
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56 views

Identical particles: Why only two possibilities?

Given two identical particles, Wikipedia says that the wavefunction of a combined system where the first particle is in state $|n_1\rangle$ and the other one is in $|n_2\rangle$ is $|\psi\rangle=|n_1\...
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87 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 ...
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Is there a proof that space expanding produces observers at all points that see what we see?

I know that galaxies are moving away from us, and so can see that it's intuitive that if space was expanding, then the astronomical observations from Earth would be the same as at all other points in ...
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206 views

Highest symmetric non-maximally symmetric spacetime

What is the highest number of symmetries (Killing vectors) that a (4-dimensional) spacetime can have without being maximally symmetric? From what I can see, it seems to be 7 (which includes the ...
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5answers
163 views

Flux of $E$ through the shaded side

A charge $q$ sits at the back corner of a cube, as shown in Figure. What is the flux of $E$ through the shaded side? One of the solution stated that. Looking at the figure, we notice two things: ...
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197 views

Permutation symmetry - a continuous symmetry?

From quantum mechanics it is known that permutation between identical particles does not change the Hamiltonian. Assuming that the quantum system consists of a very high number of particles such that ...
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66 views

Using the Mirror Rule to determine the magnetic field of an infinite slab

Consider a slab infinite in the y and z direction but with finite width W in the x direction. Current flows in the (+y) direction. I'm supposed to use the "mirror rule" to show that at a point in the (...
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189 views

Why does physics have so many symmetries?

I have just found out that in order to modify mass in his special theory of relativity, Einstein assumed that energy and momentum are always conserved.$^\dagger$ I think surely there are other ways to ...
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121 views

Canonical spinors from gauge transformations

In this 2006 paper, http://arxiv.org/abs/hep-th/0610128, there is the concept of gauge transformation and how was it employed that I do not fully understand. Note, what will be talked about below is ...
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73 views

What is meant by invariant under change of coordinates **to first order**?

I am studying elementary Lagrangian mechanics, and I'm a bit confused about the what's meant by invariance of the Lagrangian under change of coordinates to first order. More specifically, Noether's ...
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Formulating the Lagrangian in terms of invariant quantities

Consider a closed system consisting of $N$ point particles, whose Lagrangian is given in the standard way, by the total kinetic energy minus the potential energy: $\mathcal{L}(\dot{q},q):= T(\dot{q}) -...
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How is translational symmetry related to Fourier decomposition?

The book (The Cosmic Microwave Background By Ruth Durrer) about cosmological perturbations says that because of translational symmetry of the background at a constant time, we can decompose our ...
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Noether's theorem: meaning of transformation of coordinates

I have a question regarding Noether's theorem. In our introductory QFT class (which is based on the book by Michele Maggiore) we have derived the Noether currents in the same form as displayed in this ...
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1answer
47 views

Are the mass matrices the same if Higgs corresponding to different Cartan generators get a vev?

I'm trying to understand what happens when a Higgs field in the adjoint representation of a given gauge group gets a vacuum expecation value (vev). Normally, the fermions do not couple to adjoint ...
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Target Space Lorentz Invariance vs. World Sheet Weyl Invariance

The Polyakov action, $S\sim \int d^2\sigma\sqrt{\gamma}\, \gamma_{ab}\partial^a X^\mu \partial ^b X_\mu$, has the well known classical symmetries of world sheet diffeomorphism invariance, world ...
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250 views

Hermitian conjugate of an antiunitary transformation

In quantum mechanics, one often considers symmetry transformations which are defined in terms of operators which do not change the norm of states in the Hilbert space. For the Wigner's theorem, this ...
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140 views

From Noether's theorem to canonical Energy-Momentum tensor using translations

In this text that I am reading it says that the transformation $\delta \phi(x)$ is a symmetry if the Lagrangian changes by a total derivative: $$\delta \mathcal{L}= \partial_{\mu}F^{\mu} . $$ From ...
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49 views

How can crystal symmetry operations be used to reduce the number of unique properties of a solid?

Can anyone please give an example or a reference which shows how crystal point groups and symmetry operations can be used to reduce the number of parameters describing the property of a crystal, ...
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Resource for (String) Symmetry Breaking in Terms of Roots and Weights?

I'm currently searching, for quite a while now, for a paper/book that discusses symmetry breaking in terms of roots and weights. Any suggestions would be much appreciated!
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Symmetry considerations in Plane Poiseuille Flow

I'm taking a first course on fluid dynamics, and I have this (sort of) conceptual question that's been nagging me for a moment now. I can completely follow the mathematics behind the derivation of the ...
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4answers
1k views

Why, when and where is Gauss's law applicable?

Why is it said that Gauss's Law is mainly applicable for symmetric surfaces/bodies? Why not for asymmetric surfaces? I want a logical explanation! BTW my teacher said that Gauss's law is ...
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110 views

Parity of $n$-photon system

The $C$-parity (charge conjugation) of an $n$-photon system is given by $(-1)^n$. If I'm not totally wrong, the intrinsic parity of a photon is $(-1)$. What is the parity $P$ of a system of $n$ ...
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190 views

Path of light as it travels between two black holes

What would happen to light passing through a narrow space between the event horizons of two equal-mass black holes? Would it deviate or follow a straight path?
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130 views

Symmetry of extrinsic curvature tensor

I am trying to solve following problem: In a spacetime of signature (+, −, −, −), let $$ u^au_a = 1, \quad A_{ab} = \nabla_cu_dh^c_{\; a}h^d_{\; b}, \quad h_{ab} = g_{ab} - u_au_b $$ Show that ...
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Degenaracy in mass of $8$ and $27$ reps of $SU(3)$ in Coleman's Aspects of Symmetry [closed]

In Coleman's Aspect of symmetry he proposes an amusing problem in the first chapter. It asks us to consider a set of eight pseudo-scalar fields transforming in the adjoint representation of $SU(3)$. ...
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79 views

Symmetry axis and products of inertia

So if we have an object that has a symmetry axis let us say $z$-axis is a symmetry axis does this mean that the product of inertia $I_{zx} = I_{zy} = 0$? And if that is true why is it true ...
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216 views

Does isotropy imply homogeneity?

This question comes from exercise 27.1 in Gravitation by Misner, Thorne and Wheeler. They required the following: Use elementary thought experiments to show that isotropy of the universe implies ...