Questions tagged [invariants]

This tag is for questions relating to invariant, a property of a system which remains unchanged under some transformation. In physics, invariance is related to conservation laws.

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Contradicting Changes in a Lagrangian under Transformation

The change in a Lagrangian with no explicit time dependence $L(\mathbf{q},\mathbf{\dot q})$ can be written using the chain rule: $$δL = \frac{\partial L}{\partial \mathbf{q}}\cdot δ\mathbf{q} + \frac{...
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How to simplify three body decay phase space in terms of invariant mass $q^2$?

We know three body phase space is written as: $$\mathrm d\pi^3 = \frac1{2M_1}\frac{1}{(2\pi)^5}\int\frac{d^3 p_2}{2E_2} \int\frac{d^3 p_3}{2E_3} \int \frac{d^3 p_4}{2E_4} \delta^4(p_1-p_2-p_3-p_4)$$ ...
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Argument of a scalar function to be invariant under Lorentz transformations

I'm trying to prove that a Lorentz scalar object $\rho(k)$ which is a function of a cuadri-vector $k^{\mu}$ can only have a $k^2$ dependency in the argument. I can imagine that this object has to ...
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Relativity Proof with Defining interval [duplicate]

I have started reading Landau & Lifshitz Vol. 2 (fields theory) and I've got confused about something I read. to prove the Lorentz transform, it defines interval: $$ds^{2} = c^{2} dt^{2} - dr^{2}\...
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The confusion over the invariance of the correlation function and the mutually local field in the CFT

Consider the correlation function $$\langle \Pi_{i=1}^n V_i(z_i,\bar z_i) \rangle$$ such as $$\langle V_1(z_1) V_2(z_2) \rangle,(z_1>z_2)$$ by position the $z_i$ correctly, the exchange of the ...
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3 answers
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How do you prove $d\tau = dt/\gamma$ is a Lorentz invariant? [closed]

How do you prove $d\tau = dt/\gamma$ is a Lorentz invariant?
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Spin as Poincaré invariant label

I was thinking about how we construct unitary representations for the Poincaré group in the case of massive particles. We move to a frame where the particle is at rest, and here the little group that ...
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Help me understand the CPT signatures of different physical quantities

I am an engineer and not a trained physicist, but still I am fascinated by symmetries and I want to understand them. Specifically, in a paper ¹ I found the below table which covers a large number of ...
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Is physical energy is invariant under reflection?

I am wondering whether the energy landscape of a physical system (in particular a molecular conformation) can be considered invariant under reflection of the 3D space. My understanding is that some ...
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Invariant lagrangian under $SU(N)$

I know from group theory that $ \bar{R} \otimes R \otimes A= 1 \oplus \dots$ where $A$ is the adjoint representation. My question is how to build the singlet, I thought a generalization of Rodriguez's ...
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Are one-dimensional tensors of arbitrary rank just scalars?

Consider a tensor of arbitrary rank (2 for this case) $A_{ij}$, and dimension one. Granted there are two indices to specify a component, but since each index can only take one value, there is only one ...
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Reaction-Threshold Energy

I have this exercise: "Given the following reaction : $K^+ + n \rightarrow \pi^+ + \Lambda $ find : threshold energy of $K^+$, $E_{Kmin}$, to make the reaction happen. $E_{K^+}$ and $P_{\pi^+}$...
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Decay and Invariant mass [closed]

Consider the following decay : \begin{equation} K^+\longrightarrow \pi^0 + e^+ + l^0 \end{equation} where $l^0$ is a neutral lepton. What kind of particle is $l^0$? For $K^+$ , if it's at rest, ...
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Properties of the "volume element" in momentum space in relativity

I am reading Landau's The Classical Theory of Fields. On page 30, section 10, the last paragraph reads: To solve the problem, we first determine the properties of the "volume element" $...
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Completely antisymmetric unit tensor of fourth rank in different 4D coordinate systems [duplicate]

I am reading Landau's Classical Theory of Fields. On page 18, it is said that the completely antisymmetric unit tensor of fourth rank $\varepsilon^{iklm}$ is defined as the same in all coordinate ...
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Conformal Casimir as Differential Operator

my question regards equation (165) of [1], namely, how to write the conformal Casimir as a differential operator in the "usual" $z,\bar{z}$ coordinates. If one inspects the definition of the ...
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Lagrangian density transformation

In a calculation of a Lagrangian density $$ \mathcal{L}=\bar{\psi}\left(i \gamma^{\mu} \partial_{\mu}-m+i \gamma_{5} m^{\prime}\right) \psi. $$ In order to see if it is invariant or not with the ...
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Lowest kinetic energy of particle for which reaction is possible (invariant mass) [closed]

Consider the following reaction between a moving proton and a stationary proton $$p + p \rightarrow p + p + \pi^0 + \pi^0 $$ Find the lowest kinetic energy (in the labsystem of reference) for which ...
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Why this momentum identity is true? [duplicate]

In this part has this identity and I could not understand, why factor 4, for me $$k^{\mu}k^{\nu}=g^{\mu\nu}k²$$ Since $$k²=k_{\mu}k^{\mu}.$$ Unless the notation is the same but the product is ...
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Can the Lorentz transformations (or space-like interval invariance ) be demonstrated without using two spatial dimensions?

My question is at the end. Warning: my use of proper time $\Delta\tau$ and proper distance $\Delta s$ are a bit tricky in the following. I am resolving arbitrary intervals in the $\bar{\mathcal{S}}$ ...
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Casimir conformal generator of $SO(d+1,1)$

The purpose of this post is to ask the help of derivation of equation 2.8 of https://arxiv.org/abs/2106.10822 Let $P_i$ be a point on the conformal boundary of $AdS_{d+1}$ and $Z_i$ be a polarization ...
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Intuition about non-invariance of the Hamiltonian in canonical transformation

Suppose $q={\{q_i\}}_{i=1}^n$ is the set of generalized coordinates of a dynamical system. $L(q,\dot q,t)$ is the Lagrangian of the system. Now we make coordinate transformation $Q_i=Q_i(q,t)$. Then ...
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Invariant symbol, group representation

I have a question regarding the following passage from Srednicki's QFT book (p. 415) (https://web.physics.ucsb.edu/~mark/qft.html). Notations are $R$ = some representation of a lie group, $\bar{R}$ is ...
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What are the physical reasons for considering the Maxwell invariant ${\cal G}=E.B$ in the action?

As I know, the Maxwell invariant ${\cal G}=E.B$ is the fundamental invariant such as ${\cal F}=\frac{1}{2}\left( {{{\bf{B}}^2} - {{\bf{E}}^2}} \right)$ that can be used to construct all possible ...
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Definition of invariance of a QM operator under a transformation

In the Sakurai book "Modern quantum mechanics" (pg. 263) an operator $S$ is said to be invariant under a unitary transformation $T$ if: $$T^\dagger S T = S.$$ Where that definition come from?...
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Decomposition of product of two antisymmetric Lorentz tensors

Suppose I have a tensor $A_{\mu\nu}$ in the $(3,1)\oplus (1,3)$ representation of the Lorentz group where $(a,b) =(2s_a+1,2s_b+1)$. I was wondering on how to decompose explictly in terms of tensors ...
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Why do we have to divide strain engineering components to use them in the strain tensor?

I understand the reason of dividing them is in order to correctly perform coordinate transformations on strain tensors. But it feels to me as if we are tricking ourselves. Strain tensor should ...
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3 votes
1 answer
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Adiabatic Invariance and Magnetic Moment

In plasma physics, particularly, in magnetic mirrors, we have a very well-known result, which is the invariance of the magnetic moment of the particles trapped in the mirror. I've saw in some notes ...
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Physical properties or invariants encoded by the combination of metric tensor/coordinate system

In this answer it is nicely demonstrated that the numerical values $g_{\mu\nu}$ of the metric tensor depend on the coordinate system chosen or, put another way, ugly mathematical expressions can be &...
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Vacuum expectation value of vector proper fields always zero

Let $\phi$ a non-scalar vector field. Why the Lorentz invariance of vacuum expectation value has as consequence that the vacuum expectation value $v=\langle 0|\phi(x)|0\rangle$ should be zero? I ...
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What does it mean for a field theory to be invariant?

In this paper A. N. Schellekens, Conformal field theory p.8 they mention the following If a field theory has a conserved, traceless energy momentum tensor, it is invariant both under general ...
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1 answer
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Causality: Invariant Under Lorentz Transformation

I'll begin by stating that I have only studied a very small bit of special relativity: a couple of lectures or so around the end of the Physics course I took, intended just to "give us a taste&...
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Different Casimirs and Casimirs of $E_6$

I am a bit confused by the notion of Casimirs (maybe it is related to terminology). In the simplest example of $su(2)$ with generators $L_i$, we get the Casimir operator $$ L^2=\sum_i L_i^2$$ ...
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3 votes
5 answers
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What is the intuition behind the spacetime interval?

In an article that I am currently reading (under the Lorentz Invariants sub-heading), it explains that, just as the distance between two points on a Cartesian plane are obviously invariant of the ...
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What are the invariant structures in special relativity?

What geometric structures in Minkowski spacetime are Lorentz invariant?
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What do these Casimir invariants of the Galilean group physically represent?

There exist Casimir invariants of the Galilean group which commute with all the generators of the group. They are, of course, Galilean scalars (i.e., scalars under space and time translations, ...
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Can we write the mass $M$, a Casimir invariant of the Galilean group, as a function of its generators?

According to Wikipedia, the mass $M$ is one of the Casimir invariants of the Galilean group. Casimir invariants of a group are made out of the generators, and they commute with all the generators of ...
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What physical properties are invariant under relativistic transformation?

Most of the familiar physical properties vary according to the relativistic observer's reference frame - speed, mass, energy, time, length, etc. Which properties remain invariant, so everybody will ...
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Landau & Lifshitz "Classical Field theory" argument for invariance of $ds^2$

In Landau & Lifshitz's "classical field theory", chapter 1, I'm getting mad with an argument used to derive the $ds^2$invariance from Einstein's postulate of the invariance of the speed ...
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Co-ordinate invariance in Lagrangian form of equations

I have read that in his Mecanique Analytique [1788], Lagrange sought a “coordinate invariant expression for mass times acceleration”. The discussion regarding this is given in 'Marsden and Ratiu [15, ...
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What are some of the uses of the invariant spacetime interval? [closed]

I am currently researching special relativity and I have come across the invariant interval. So far, I know that all observers will measure the same interval regardless of position or velocity, and ...
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6 votes
1 answer
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How can you make mechanics invariant under inversion of length? [closed]

I read that in some string theories they are invariant under a change of distance from $r$ to $1/r$. I wondered what is the simplest theory or lagrangian in 4 dimensions that could have this property. ...
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Is the speed of light through a medium invariant, just like the speed of light through vacuum is invariant? Also, do time dilation etc. still occur?

So, we know the speed of light through a vacuum is $c$. Let us say that both our observers are moving past each other at speed $v$ in a medium in which the speed of light is $c'$. So, does the usual ...
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What are the physical state invariants of loop quantum gravity?

What are the "physical state" invariants for Loop Quantum Gravity? The Wikipedia page talks about "physical states" being invariant, diffeomorphism invariance, "quantities ...
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Coordinate invariance in Physics

Let us consider a classical field theory on flat background spacetime. The action is $$S[\Phi] = \int d^nx \mathcal{L}(\Phi,\partial_\mu\Phi).$$ Why shouldn't this action be independent of the chosen ...
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Analyzing the Schrodinger equation under Lorentz and Galilean transformation

I'm trying to see what happens to the Schrodinger equation under Lorentz and Galilean transformation. So I assumed a free Schrödinger wave with the form: $$\psi = A e^{i(\omega t - kx)} ~.$$ A Galilei ...
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1 vote
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Lattice Gas Automata and Galilean Invariance

I have been studying Lattice Gas Automata methods (also this), and every time I read up on their drawbacks, I see that they are not Galilean invariant and that the simulations have statistical noise. ...
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2 votes
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Can we derive the momentum of photon from special relativity?

I don't really have strong backgrounds studying quantum physics, but I did learn special and general relativity, and I have now a question how to get the momentum of photon. For my understanding, ...
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Is mass still a scalar in special relativity?

In Euclidean space is the space of classical Mechanics, A scalar is the same for all observers that are to say remain invariant under the change of coordinate systems. A Vector $\mathbf{V}$ is a ...
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1 vote
2 answers
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Is the rest mass of a system invariant even if it is a function of time?

If we imagine a system of particles, and we consider only $N$ of them, then the rest mass of these $N$ particles is given by $$\left(\sum_{i=1}^N P_i\right)^2=P_T^2=m^2$$ Where $P_i$ is the 4-momentum ...
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