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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Why is it desirable to have a symmetry to make cosmological constant zero?

It is sometimes stated that absence of a symmetry to make cosmological constant zero is a problem. But observed value of dark energy is very small and non-zero. So why is it desirable to have a ...
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1answer
167 views

Relationship between local and global scaling (Weyl) symmetry

Theorem 5.1 on page 80 of this paper says that Assuming that the matter fields satisfy their equations of motion, the matter field action is locally Weyl invariant if and only if the corresponding ...
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2answers
532 views

How to apply Noether's theorem

Say I have a point transformation: $$x' ~=~ (1 +\epsilon)x,$$ $$t' ~=~ (1 +\epsilon)^2t,$$ and Lagrangian $$ L ~=~ \frac{1}{2}m\dot{x}^2 - \frac{\alpha}{x^2}.$$ How do I go out about showing ...
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98 views

Poynting vector and Rindler flux under time inversion

This question is about some reply by John Baez on sci.physics.research the post is this: https://groups.google.com/d/msg/sci.physics.research/F6x5GkFt0ic/fxsfuNl9d8gJ the article he is talking about ...
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164 views

Why Must Conserved Currents of Lorentz Symmetry Satisfy the Lorentz Algebra

I've seen it written many times that the commutation relation $[M^{I-},M^{J-}]=0$ is required for Lorentz invariance in the light cone gauge quantisation of the bosonic string. This follows ...
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369 views

CPT Violation and Symmetry / Conservation Laws

Ok, so I remember reading that every conservation law has a corresponding symmetry (i.e. conservation of momentum is translational symmetry, conservation of angular momentum is rotational symmetry). ...
3
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1answer
176 views

Question on Section 9.1.3 in “Conformal Field Theory” by Philippe Di Francesco et. al

Question on Section 9.1.3 in "Conformal Field Theory" by Philippe Di Francesco et. al. The basic idea of the Coulomb-gas formalism is to place a background charge in the system, making the $U(1)$ ...
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853 views

Schrödinger function: Separable wave function with even potential function of x

I have done the Problem 2.1 in Griffiths' quantum mechanics, and it seems not making sense to me. What if the wave function isn't symmetric at all? Then obviously the proof doesn't work. The ...
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0answers
89 views

Division algebras $(\mathbb{R,C,H,O})$ and discrete symmetry [closed]

I once saw a statement about the relation between division algebra(which means you can define a division in this algebra, there is a theorem saying we only have 4 kinds of division algebra, real R, ...
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2answers
721 views

What is the role of the vacuum expectation value in symmetry breaking and the generation of mass?

Consider a theory of one complex scalar field with the following Lagrangian. $$ \mathcal{L}=\partial _\mu \phi ^*\partial ^\mu \phi +\mu ^2\phi ^*\phi -\frac{\lambda}{2}(\phi ^*\phi )^2. $$ The ...
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2answers
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Is this Landau's other critical phenomena mistake?

There was an old argument by Landau that while the liquid gas transition can have a critical point, the solid-liquid transition cannot. This argument says that the solid breaks translational symmetry, ...
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147 views

Symmetries of spacetime and objects over it

I guess according to mathematical didactic, we first think of spacetime as a set and we reason about elements of its topology and then it's furthermore equipped with a metric. Appearently it is this ...
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242 views

Symmetry and overlapping of ground states

In a quantum mechanics, there is the following formula to derive the zero energy $E_0$ of a perturbed Hamiltonian $$H = H_0 + V$$ knowing the zero energy $W_0$ of the free Hamiltonian $H_0$: $$E_0 = ...
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153 views

Why does renormalization need an unbroken symmetry?

Common wisdom is that for a QFT to be renormalizable it must be invariant under a symmetry transformation. Why does renormalization need an unbroken symmetry? Which is the first publication that ...
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1answer
54 views

Testing covariance of an expression?

This is something I've been unsure of for a while but still don't quite get. How does one tell whether an expression (e.g. the Dirac equation) is covariant or not? I get it for a single tensor, but ...
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3k views

Why are snowflakes symmetrical?

The title says it all. Why are snowflakes symmetrical in shape and not a mush of ice? Is it a property of water freezing or what? Does anyone care to explain it to me? I'm intrigued by this and ...
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1answer
393 views

What happens to the Lagrangian of the Dirac theory under charge conjugation?

Consider a charge conjugation operator which acts on the Dirac field($\psi$) as $$\psi_{C} \equiv \mathcal{C}\psi\mathcal{C}^{-1} = C\gamma_{0}^{T}\psi^{*}$$ Just as we can operate the parity operator ...
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105 views

Dilatations in non-relativistic QM and operator tranformation

I was looking at a QM textbook exercise dealing with dilatations, the transformations are $x \rightarrow x' = \lambda x$ transforming $|\psi\rangle$ into $|\psi'\rangle = ...
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79 views

How to deal with crossing duality and modular invariance in string field theory?

An answer I gave elsewhere. Some cases to ponder over. A closed string splits into two closed strings, which then merge again into a single closed string. The overall string worldsheet has ...
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1answer
76 views

How can we have massive states of strings and CFT on the string worldsheet at the same time?

Ok, so we can have conformal invariance on a string world sheet. However, it is well known that to preserve conformal symmetry we require states to be massless. So how is it that string theories ...
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2answers
262 views

Scale invariance symmetry as a simple argument in an electrostatics problem

In the comments to this post, it was hinted that proving that the force acting on a charge at a vertical distance from a uniformly charged plane is independent of that distance can be done by ...
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2answers
339 views

Invariance of Maxwell's Equations under inverting variables - Reference and use

Some months ago, an ArXiv paper mentioned in passing that Maxwell's Equations were invariant under reciprocating the variables, or at least this results in a dual set of Maxwell Equations. (Actually I ...
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218 views

Breaking of conformal symmetry

I am wondering something about the breaking of conformal symmetry: I know that it can be broken at the quantum level, anomalously, but I never encountered or heard about a model where it is broken "à ...
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1answer
507 views

Even and Odd States of a 1D finite potential well

Is it possible for a particle trapped in a 1D finite potential well to evolve from a even state to an odd state and vice-versa? Why?
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631 views

Conservation Laws and Symmetries

Usually, in Quantum Mechanics, an observable is an operator on the space of the possible quantum states (labelled as $|\psi\rangle$). If this quantity is conserved, in the meaning that the associated ...
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398 views

The Asymmetry between Real and Imaginary in the three Pauli Spin Matrices

The Pauli spin matrices $$ \sigma_1 ~=~ (\begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix}), \qquad\qquad \sigma_2 ~=~ (\begin{smallmatrix} 0 & -i \\ i & 0 ...
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3answers
515 views

What is the difference between manifest Lorentz invariance and canonical Lorentz invariance?

I often read that the Lorentz symmetry is manifest in the path integral formulation but is not in the canonical quantization - what does this really mean?
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93 views

Excitations implied by symmetries

I read that in condensed matter field theory a symmetry implies not only a conserved current (through the well-known Noether theorem) but some kind of "low energy excitation". I am familiar with the ...
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3answers
223 views

Representation of phase in quantum mechanics

[Note: My discussion of the three answers can be found just after the question.] Imagine three points in space that differ only by a phase angle of "something" (what doesn't really matter). One way ...
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637 views

Lorentz Invariance of Maxwell Equations

I am curious to see a simple demonstration of how special relativity leads to Lorentz Invariance of the Maxwell Equations. Differential form will suffice.
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2answers
363 views

Symmetries, Generators, Commutators and Observables

I'm learning about generators and conservation laws and have derived the equation (1) $$[Q,A]=-i\hbar f(A)$$ which is satisfied by the observable generator $Q$ for a transformation group with ...
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1answer
321 views

Why are all observable gauge theories not vector-like?

Why are all observable gauge theories not vector-like? Will this imply that the electron and/or fermions do not have mass? How is this issue resolved? Background: The Standard Model is a ...
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1answer
447 views

Time reversal symmetry and T^2 = -1

I'm a mathematician interested in abstract QFT. I'm trying to undersand why, under certain (all?) circumstances, we must have $T^2 = -1$ rather than $T^2 = +1$, where $T$ is the time reversal ...
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1answer
485 views

Time reversal symmetry and T^2 = -1

I'm a mathematician interested in abstract QFT. I'm trying to undersand why, under certain (all?) circumstances, we must have $T^2 = -1$ rather than $T^2 = +1$, where $T$ is the time reversal ...
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1answer
152 views

Searching the point group of symmetry

I am engaged in the field of quantum-chemical calculations using programs written by myself. I have found out that I have a problem in finding the point group symmetry of the molecule. The first idea ...
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1answer
162 views

transformations with commutators and anticommutators that generate displacements

is well known that composition of point reflections generate pure displacements. This implies that the commutator of two point reflections will be a pure displacement. Are there similar elemental ...
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2k views

Spontaneous Time Reversal Symmetry Breaking?

It is known that you can break P spontaneously--- look at any chiral molecule for an example. Spontaneous T breaking is harder for me to visualize. Is there a well known condensed matter system which ...
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142 views

Does turbulence violate Galilean relativity?

Fluid flows become turbulent beyond a certain velocity. The velocity is almost always with respect to a fixed boundary. However, an observer in a frame of reference travelling with the fluid will also ...
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1answer
275 views

Lepton Number Conservation

What is the global symmetry of the electroweak Lagrangian that gives rise to lepton number conservation? As I understand it, electric charge is some linear combination of the conserved quantities ...
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1answer
136 views

Request for Reference: BRST formalism/transformations

Could anyone please suggest a very basic paper/reference/literature on BRST symmetry/formalism that requires rudimentary knowledge of Dirac's method for dealing with constrained systems and generation ...
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2answers
259 views

Scalar potential, vector potential, and spinor potnetial

In Particle Physics, I've seen Scalar potentials which look like this $$ V = a \Phi^2 + b \Phi^4$$ $\Phi$ is scalar (a number). What about vector potentials, and spinor potentials? How are they ...
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1answer
389 views

Spontaneous symmetry breaking and 't Hooft and Polyakov monopoles

What is spontaneous symmetry breaking from a classical point of view. Could you give some examples, using classical systems.I am studying about the 't Hooft and Polyakov magnetic monopoles solutions, ...
2
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1answer
321 views

What is the Lie algebra of the Galilean group and what is the structure of it?

I read Freeman Dyson's article Missed Opportunities, in which he talked about the mathematical attractiveness of the Lorenz group compared to the Galilean group. I am reading Florian Scheck's book on ...
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94 views

Rotationally invariant body and principal axis

Suppose a rigid body is invariant under a rotation around an axis $\mathsf{A}$ by a given angle $0 \leq \alpha_0 < 2\pi$ (and also every multiple of $\alpha_0$). Is it true that in this case the ...
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1answer
295 views

Lorentz invariance of a frequency- and wavelength- dependent dielectric tensor

Suppose we have a material described by a dielectric tensor $\bar{\epsilon}$. In frequency domain, this tensor depends on the wave frequency $\omega$ and the wave vector $\vec{k}$. Clearly not all ...
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1answer
1k views

Understanding units and the units of the derivative operator

Suppose that $f$ is a function from unit $A$ to $B$, then what is the unit of $f'(x)$?. We can do $f'(x)\Delta x$ to get an estimate of $f(x + \Delta x)$. Since the latter has unit $B$, so has the ...
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432 views

Why do humans have bilateral symmetry? [closed]

About the eyes I know that it requires for gauging distance as in Modern 3D cameras have two sensors. And two ears for sound source localization using differences in levels and timing (But not yet two ...
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456 views

Wigner-Eckart projection theorem

I'm following the proof of Wigner-Eckart projection theorem which states that: $$\langle \bf{A} \rangle ~=~ \frac{\langle \bf{A} \cdot \bf{J} \rangle}{\langle {\bf{J}}^2 \rangle} \langle \bf{J} ...
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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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221 views

Goldstone's theorem and massless modes for $\phi^4$ theory

Consider a scalar field doublet $(\phi_1, \phi_2)$ with a Mexican hat potential $$V~=~\lambda (\phi_1^2+\phi_2^2-a^2)^2.$$ When $a=0$ this is a quartic potential and the symmetry is not ...