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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122 views

Diffeomorphism group vs. $GL(4,\mathbb{R})$ in General Relativity

I am quite confused with the groups Diff$(M)$ and $GL(4,\mathbb{R})$ in the context of general relativity. I understand that the symmetries of GR are the transformations that leave the equations ...
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3answers
3k views

Noether's current expression in Peskin and Schroeder

In the second chapter of Peskin and Schroeder, An Introduction to Quantum Field Theory, it is said that the action is invariant if the Lagrangian density changes by a four-divergence. But if we ...
2
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0answers
107 views

Noether's theorem and translations

I'm a bit confused about Noether's theorem (or about calculus of variations in general) when it comes to the translational symmetry $x^\mu\mapsto {x'}^\mu=x^\mu-a^\mu$. My professor just wrote that if ...
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3answers
476 views

When do phase space functions' Poisson brackets inherit the Lie algebra structure of a symmetry?

I've seen several examples of phase space functions whose Poisson brackets (or Dirac brackets) have the same algebra as the Lie algebra of some symmetry. For example, for plain old particle motion in ...
4
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1answer
229 views

Symmetry factor for Feynman diagrams in $\phi^4$-theory for $n$-points Green function

I'm working with two theories. Theory A: $H_{int} =\int d^3x \frac{Mg}{2}\phi\varphi^2$ Theory B: $\phi^4$-interaction: $H_{int} = \int d^3 x \frac{\lambda}{4!}\phi(x)^4$ Where $M$ is the mass ...
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1answer
68 views

How is it possible to vary time without affect the coordinates or their derivatives?

In the context of Noether's theorem , the Hamiltonian is the constant of motion associated with the time-translational invariance of the Lagrangian. Time-translational invariance is equivalent to the ...
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0answers
24 views

Global and local symmetry for Isospin/Strangeness etc

Why some symmetries $ \big[SU(3),SU(2)$ and $U(1)\big]$ of the Standard Model are local, and some others remain global, like Isospin and Strangeness. Is there a fundumental reason for that? Doesn't it ...
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45 views

Do we always have particle-hole symmetry?

If we write the BdG equation for any model, and double the degrees of freedom (e.g. 4N*4N matrix for a N site chain), then we are guaranteed the particle-hole symmetry. Is there any constraints to do ...
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0answers
65 views

Symmetry of interaction lagrangian and symmetry of full lagrangian

Suppose we have lagrangian $$ \tag 1 L = \frac{\theta}{f_{\gamma}}F_{EM}\tilde{F}_{EM} +\frac{1}{2}(\partial_{\mu}\theta)^2 - \frac{1}{2}m_{\theta}^2\theta^2 + L_{SM}, $$ where $\tilde{F}_{EM}$ ...
2
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1answer
51 views

physical meaning of major symmetry of the stiffness tensor

What happens if a stiffness tensor does not have the "major symmetry" $C_{ijkl}=C_{klij}$? Background: In linear elasticity (generalising Hooke's law from a spring to a continuous medium), the ...
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2answers
59 views

Is this an error?

The teacher wrote the following: There is a dot missing where the green arrow is, right? After applying Euler's theorem, the term in brackets becomes $x_j$, but we need it to be $\dot{x_j}$, don't ...
2
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1answer
351 views

What's a good book for an advanced undergraduate/early graduate student to learn about symmetry, conservation and Noether's theorems?

What's a good book (or other resource) for an advanced undergraduate/early graduate student to learn about symmetry, conservation laws and Noether's theorems? Neuenschwander's book has a scary review ...
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1answer
104 views

Are all maximally symmetric spacetimes constant curvature spacetimes?

A $d$ dimensional maximally symmetric spacetime is a spacetime with the maximum allowed number of Killing vectors. This number is $\frac{d(d+1)}{2}$. Constant curvature spacetimes are spacetimes ...
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1answer
89 views

Lagrangian vs Hamiltonian and symmetry of a theory

It is said that since the path-integral formulation of quantum mechanics/or quantum field theory uses the Lagrangian rather than the Hamiltonian, as the fundamental quantity, it preserves all the ...
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4answers
873 views

Are the principles of space-time homogeneity and Isotropy independent of one another?

Einstein in deriving the Lorentz transformations, used the principles of space-time homogeneity and Isotropy. Does space-time isotropy follow from space-time homogeneity or are they completely ...
3
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1answer
210 views

Particle-Hole transformation in Superconductor

The Bogoliubov - de Gennes equation has a emergent particle-hole symmetry: $$ \mathcal{P}H\mathcal{P}^{\dagger} = -H\text{.} $$ My question is now what happen to the Nambu spinor: $$ ...
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49 views

Superpotential Symmetry

Superpotential in general has the form $W=a_n\Phi^n$. If I require that my superpotential should be invariant under the following global transformation, $\delta \Phi=i\epsilon \Phi$ and $\delta ...
2
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1answer
132 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 ...
4
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1answer
87 views

$SO(N)$ symmetric theory of $N$ real scalar fields, why do charges have correct commutation relations of generators?

Consider an $SO(N)$ symmetric theory of $N$ real scalar fields,$$\mathcal{L} = {1\over2} \partial_\mu \Phi^a \partial^\mu \Phi^a - {1\over2} m^2 \Phi^a \Phi^a - {1\over4} \lambda(\Phi^a ...
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1answer
106 views

Gauge the symmetry $φ \to φ + a(x)$ for a free massless real scalar field

How does one alter the Lagrangian density for a real scalar field $$\frac{∂_μφ∂^μφ}{2}$$ such that is will be invariant under the gauge transformation $φ → φ + a(x)$? For a complex scalar field ...
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1answer
153 views

Behavior of the Electric- and Magnetic-field under time reversal and parity

The behavior of the electric- $\mathbf{E}$ and the magnetic-field $\mathbf{B}$ und time reversal and parity can be calculated in different ways. My first solution is to study the transformation ...
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2answers
186 views

Conservation of energy and Killing-field

In general relativity we have no general conservation of energy and momentum. But if there exists a Killing-field we can show that this leads to a symmetry in spacetime and so to a conserved quantity. ...
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5answers
453 views

Is Parity really violated? (Even though neutrinos are massive)

The weak force couples only to left-chiral fields, which is expressed mathematically by a chiral projection operator $P_L = \frac{1-\gamma_5}{2}$ in the corresponding coupling terms in the Lagrangian. ...
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56 views

Combinatorics of fourth order feynman diagram

I am trying to calculate how many different forth order feynman loop diagrams I can produce. I know that for 2nd order it is 6x3x2 thus 3! since you start with 3 lines coming out of each vertex so 6 ...
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2answers
194 views

Is there something more to Noether's theorem?

From the definition of Lagrangian mechanics, Noether's theorem shows that conservation of momentum and energy comes from invariance vs time and space. Is the reverse true? Are Lagrangian mechanics ...
8
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1answer
407 views

Invariance of action $\Rightarrow$ covariance of field equations?

Invariance of action $\Rightarrow$ covariance of field equations? Is this statement true? I have only seen examples of this, like the invariance of Electromagnetic action under Lorentz ...
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48 views

Anisotropy Axis

I am trying to identify the anisotropy axis of a patterned two-dimensional surface, such as one with parallel sets of stripes...is the anisotropy axis parallel or perpendicular to the stripes? ...
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0answers
36 views

What are gyrotropic crystals?

I am reading a paper about spin photocurrents that talks about gyrotropic crystals, but I do not find any good explanation what gyrotropy is or how to find out whether a crystal belongs to the ...
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0answers
81 views

Non-symmetry of a lagrangian

If a transformation $\Phi \rightarrow \Phi + \alpha \partial \Phi/ \partial \alpha$ is not a symmetry of the Lagrangian, then the Noether current is no longer conserved, but rather ...
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0answers
109 views

Constructing Killing tensors from Killing vectors

Background: After reading about Carter constant and symmetries in GR, I became interested in Killing tensors. I tried reading this paper by Alan Barnes, Brian Edgar and Raffaele Rani, discussing ...
8
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3answers
406 views

Why is Planck's constant the same for all particles?

This question came to me while reading Where does de Broglie wavelength $\lambda=h/p$ for massive particles come from? This question has a nice answer that explains that wave number has be ...
13
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1answer
1k views

Do an action and its Euler-Lagrange equations have the same symmetries?

Assume a certain action $S$ with certain symmetries, from which according to the Lagrangian formalism, the equations of motion (EOM) of the system are the corresponding Euler-Lagrange equations. Can ...
0
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1answer
93 views

Gravitational force and potential in infite slab

Let's say that we have an infinite slab of height $2h$ and mass density $\rho$. Let's define $x,y$ as the axis parallel to the slab and $z$ as the perpendicular one, with $z=0$ at the middle of the ...
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1answer
465 views

Symmetry arguments for valley physics in graphene with broken inversion

I am trying to understand this paper: http://link.aps.org/doi/10.1103/PhysRevLett.99.236809 (Here is an arXiv version: http://arxiv.org/abs/0709.1274) In the introduction, they mention certain ...
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5answers
1k views

Noether Theorem and Energy conservation in classical mechanics

I have a problem deriving the conservation of energy from time translation invariance. The invariance of the Lagrangian under infinitesimal time displacements $t \rightarrow t' = t + \epsilon$ can be ...
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2answers
533 views

Killing Vectors in Schwarzschild Metric

Given the Schwarzschild metric with $(-,+,+,+)$ signature, $$ds^2=-\left(1-\frac{2M}{r}\right)dt^2+\left(1-\frac{2M}{r}\right)^{-1}+r^2(d\theta^2+\sin^2\theta\,d\phi^2)$$ the lack of dependence of ...
0
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1answer
69 views

Transformations of states in quantum mechanics

In Classical Mechanics we usually describe the possible configurations of a system by points on a smooth manifold $M$ which is the configuration manifold of the system. In that case, when we talk ...
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3answers
64 views

Examples of non-linear field symmetries?

Consider a Lagrangian theory of fields $\phi^a(x)$. Sometime such a theory posseses a symmetry (let's talk about internal symmetries for simplicity), which means that the Lagrangian is invariant under ...
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2answers
183 views

Is my understanding of Gauge Symmetries correct?

I'm currently working on a project about Symmetry Breaking for my physics bachelor. Right now I'm trying to understand Gauge Symmetries (although I guess it's not much of a symmetry). And I've been ...
5
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1answer
124 views

How to find symmetry transformations?

For a given Lagrangian $$ {\cal L} = - \frac{1}{4} F_{\mu \nu} F^{\mu\nu} + |D_{\mu} \phi|^2 -V (\phi) $$ with $\phi = \frac{1}{\sqrt{2}} (\phi^1 + i \phi^2)$, there are the infinitesimal local ...
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0answers
37 views

Advanced Quantum Mechanics (Galilean transformations)

I have been reading quantum mechanics textbook by Ballentine, and in the third chapter, he says that the eigenvalues of the transformed operator, A', must remain same as those of A. I am confused ...
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1answer
43 views

Isotropic of Inertial frame?

My understanding of isotropic is the a particular physics law remain same no matter at what direction I look at it? Now suppose in case of inertial frame, we know that its is homogeneous and ...
0
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1answer
62 views

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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0answers
21 views

Symmetry of retarded R-current correlator in $\mathcal{N}=4$ Super Yang-Mills

The retarded correlator of the R-current $J_\mu$ of $\mathcal{N}=4$ Super Yang-Mills theory is $$ C_{\mu\nu}(x-y)=-i\theta(x^0-y^0)\langle[J_\mu(x),J_\nu(y)]\rangle. $$ In this paper in eq. (2.4), I ...
23
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1answer
2k views

Emergent symmetries

As we know, spontaneous symmetry breaking(SSB) is a very important concept in physics. Loosely speaking, zero temprature SSB says that the Hamiltonian of a quantum system has some symmetry, but the ...
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Can someone give a simple expose on Coleman Mandula theorem and what Mandelstam variables are?

Can someone give a simple expose on Coleman Mandula theorem and what Mandelstam variables are? Coleman-Mandula is often cited as being the key theorem that leads us to consider Supersymmetry for ...
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26 views

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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5answers
3k views

Is the converse of Noether's first theorem true: Every conservation law has a symmetry?

Noether's (first) theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. Is the converse true: Any conservation law of a physical ...
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1answer
55 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 ...
8
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4answers
593 views

What role does “spontaneously symmetry breaking” played in the “Higgs Mechanism”?

In talking about Higgs mechanism, the first part is always some introduction to the concept of spontaneously symmetry breaking (SSB), some people saying that Higgs mechanism is the results of SSB of ...