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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A question about the Lorentz transformation of “infinitesimals”

Notations conventions: $p$ stands for the momentum (so $d^3p$ is the differential element according to which we integrate, for the $3$ space coordinates). A Lorentz transformation is denoted by $\...
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Invariant of supersymmetry?

Given two vectors in 3D superspace $(x_1^\mu,\theta_1^\alpha,\overline{\theta}_1^\alpha)$ and $(x_2^\mu,\theta_2^\alpha,\overline{\theta}_2^\alpha)$ I am trying to find a polynomial invariant under ...
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Is temperature a Lorentz scalar [duplicate]

If I see a body at temperature $T$, will I see the same temperature in another frame under a Lorentz boost. And will the internal energy of a body also remain invariant under a Lorentz boost or not..
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What are all objects that are invariant under Lorentz transformations?

I am interested in the following question: What are all objects that are invariant under Lorentz transformations? And, once a list is provided, how to justify that these are indeed ALL such objects? ...
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Finding if a quantity is a Lorentz scalar

I am new to special relativity, and I am trying to figure out if $\phi(x)=\frac{a \cdot a}{x \cdot x+a \cdot a-x^{0} a^{0}}$ is a Lorentz scalar, where $x$, and $a$ are four-vectors. Since the dot ...
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Invariance of inner product under Poincare transformation

The Poincare transformation reads, $$x\rightarrow x^\prime=\Lambda x +a $$ The scalar product is preserved under Lorentz transformation. However I do not see how it is preserved under the more general ...
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Quadratic Casimir of Higher dimensional representations

I am trying to find the quadratic Casimir of a 45 dimensional representation of ${\rm SU}(5)$. However, in many references, the dimension of the representation of ${\rm SU}(N)$ are $N, N^2-1, N(N-1)/2,...
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Interest of having 2 system of coordinates in the four-dimensional volume form

In the following expression of Lagrangian in General Relativity : $$S=\int d^{4} x \sqrt{-g}\left(\frac{R}{16 \pi G}+\mathcal{L}_{\mathrm{M}}\right)$$ I understand that we can write for example : $$c\,...
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Most fundamental reason for Newtonian KE loss being invariant in inelastic collisions

This answer to a question about why Newtonian kinetic energy is quadratic in velocity shows that if an inelastic collision's KE loss is invariant under Newtonian boosts it has to quadruple when ...
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Can energy momentum four-vectors be used to solve a relativistic kinematics problems when a centre of mass energy $\sqrt s$ is given? [closed]

This is a follow-up to this previous question. We are encouraged to use 4-vectors to solve these types of problems: Decay kinematics using 4-vectors. Here we look at the kinematics of particle decays ...
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How does time translational invariance and linearity imply exponential solutions?

I'm currently studying "Waves and Oscillation". While going through the book The Physics of waves, from page 11-12. The author has mentioned that the differential equation being linear ...
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Fourier expansions of Klein Gordon field not Lorentz invariant?

I’m working in Peskin and Schroeders book on QFT and noticed that they expanded a solution to the Klein Gordon equation in a manner that seems to me not to be be Lorentz invariant even though the ...
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Scaling invariance of fluid energy

I have read that the incompressible Navier Stokes equation is preserved by the scaling $$x',y',z'=\lambda x, \lambda y, \lambda z$$ $$t'=\lambda^2 t$$ $$u'=(1/\lambda) u$$ As I understand it, fluid ...
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Electromagnetic duality interacting with a complex scalar field

My question refers to example theory introduced in the book "Supergravity" from D.Z.Freedman & A. van Proeyen p.80. Its Lagrangian is given by $${\cal L}(Z,F) =-\frac{1}{4}(Im Z)F_{\mu\...
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Winding number as topological invariant in Su-Schrieffer-Heeger (SSH) model

I'm studying the SSH model, here's the reference. I don't get what the definition of a topological invariant is in this case. I think the important property is that the winding number cannot be ...
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Casimir operator and the BTZ gauge invariant quantities

I am a bit confused with the information that is provided by the Casimir operator. First, with my understanding, a Casimir operator is defined as, $$\Omega_\rho := \sum_{i-1}^{dim L} \rho(X_i) \circ \...
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Is $E^2-P^2=m^2$ true only for free particles?

I'm studying Friedman and Susskind's Special Relativity and Classical Field Theory and follow them in using $c=1$. They derive the above relation by first using Lagrangian of a free particle $\mathcal ...
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How to be sure that a law is invariant under Lorentz's Transformation?

For starters let's talk about Maxwell's Equations; we know that Maxwell's Equations are invariant under Lorentz's Transformation, after all this is why all the relativity business got started. To ...
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Chern-Simons (CS) theory

I have a question about Constructuion of Chern-Simon Action. In its paper "Non-commutative geometry and string field theory", Witten construct the Action of the String Field Theory inspiring ...
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How can $F_{\mu\nu}F^{\mu\nu} = 2(B^2-E^2)$ be proved? [closed]

How does $F_{\mu\nu}F^{\mu\nu} = 2(B^2-E^2)$? $$ F_{\mu\nu}=\pmatrix{ 0&E_x&E_y&E_z\\ -E_x&0&-B_z&B_y\\ -E_y&B_z&0&-B_x\\ -E_z&-B_y&B_x&0 } $$ $$ F^{\mu\...
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Fundamental invariants of the electroweak sector?

In a previous question, I asked what the matrix representation of the electroweak fields is, and I was told they are identical to the Faraday tensors, but come in a set of three ($W_i, i\in \{1,2,3\}$)...
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What does invariance of Lagrangian under a group action mean?

Let $L(q_i,\dot{q_i},t)$ be the(a?) Lagrangian of a physical system. Assume that the gen. coordinates $q_i$ transform under a certain Group G as $q_i\rightarrow q_i'=f_i(q_j,\theta_k)$ where $f_i$ are ...
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Transformation of Lagrangian and action

Consider the Lagrangian $L(q_i,\dot{q_i},t)$ for $i=1,2, ...n$. Transform (invertibly) $q_i$ to another set of generalized coordinates $s_i=s_i(q_j,t)$. Now, in a different scenario, consider ...
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Why is the Lorentz invariant integration measure for a spinor $\frac{d^3 k}{(2\pi)^3}\frac{m}{\omega}$?

I understand for a scalar field theory the integration measure is $\frac{d^3 k}{(2\pi)^3}\frac{1}{2\omega}$ because it has to satisfy the following equation $$\int \frac{d^4 k}{(2\pi)^4}\delta(\omega^...
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Angle-preserving linear transformations in 2D space for relativity

I'm watching this minutephysics video on Lorentz transformations (part starting from 2:13 and ending at 4:10). In my spacetime diagram, my worldline will be along the $ct$ axis and the worldline of an ...
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Proving that timelike and spacelike spacetime intervals are invariant across inertial frames

I'm trying to understand the justification for using the Minkowski metric. It's clear to me that it's the natural choice of metric given that spacetime separations denoted by $(-c^2\Delta t^2+\Delta x^...
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Invariance of Inner product between 4-velocity under general coordinate transformation

I know that inner product between 4-velocity is invariant under Lorentz transformation and I know that inner product between any 2 vectors under general coordinate transformation is invariant. ...
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1answer
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Proving an object is a $4$-vector given its inner product with a $4$-vector is a scalar

Theorem: Suppose $A_{\mu}$ is a $4$-vector and $B^{\mu}$ is an object with $4$ components. If $A_{\mu}B^{\mu}$ is a scalar then $B^{\mu}$ is a $4$-vector. I have been stuck on trying to prove this ...
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Invariance of Lagrangian under Poincaré group transformations implies covariant Lagrange equations? [duplicate]

I'm taking a class on classical fields and I came across a statement that I can't think about an argument to show that its true. It says that Invariance of a Lagrangian under transformations on ...
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Physical reason for defining a Lorentz transformation as one that preserves the inner product of 4-vectors?

There's a nice answer to this question: Why is the scalar product of two four-vectors Lorentz-invariant? - that explains that a Lorentz transformation is one under which the inner product of two 4-...
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Why is mass an invariant in Special Relativity?

I have read here that mass is an invariant and that it is the momentum that approaches infinity when your speed approaches the speed of light. That is why infinite energy is required to accelerate an ...
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Is it right to think of Parity as a change of basis in Dirac's Lagrangian?

I'm trying to understand CPT symmetries in the Dirac Lagrangian but, so far, I've had more questions than answers. My naive view of CPT transformations is the following (please don't doubt to correct ...
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How many Lorentz scalars are needed to characterise $n$ 4-vectors?

If I have an arbitrary function of $n$ 4-vectors $f = f(q_1^\mu, q_2^\mu, ..., q_n^\mu)$ where $q_i^\mu$ are 4-vectors, what is the least number of Lorentz scalars I would need if I needed to specify ...
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Charge is not only a scalar (or invariant) under rotation; it is also invariant for frames of reference in relative motion

I read this statement in my textbook (here) Charge is not only a scalar (or invariant) under rotation; it is also invariant for frames of reference in relative motion. I am not able to understand ...
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Difference between: 'Rest Mass' length, 'Centre of Mass' length and 'Invariant Mass' length in special relativity and particle physics?

Firstly, after reading this source, the length, $s$ of an energy-momentum four-vector is $$s=m_0c^2\quad\fbox{Rest Mass}$$ But, according to this, the length of an energy-momentum four-vector is $$...
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Is metric in GR invariant?

In SR, the central theme is the invariance of Lorentz metric, but under a particular Lorentz transformation. In GR, a metric is a solution to the Einstein’s field equation and there can be all kinds ...
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Show invariance of the inner product of $4$-velocities in different frames

In the lab frame, particle $B$ moves to the right with speed $u$, and particle $C$ moves to the left with speed $v$. In the frame of $C$, particle $B$ is seen to move to the right with speed $w$, ...
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Quartic Casimir of the 3D conformal group

I am studying the conformal group in 3 dimensions. The generators of this group are isomorphic to the generators of $SO(1,4)$. Hence two of the Casimir operators are, $$C_1=-\tfrac12J_{AB}J^{AB}$$ $$...
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Reparametrization of einbein action

I would like to show that the following action $$ \mathcal{S}=-\frac{1}{2}\int{d\tau \sqrt{-g_{\tau\tau}}\left(g^{\tau\tau}\eta_{\mu\nu}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau}+m^2\right)} $$ is ...
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Invariant Interval Interpretation

Thanks for reading. The invariant interval between two events is defined as... $S^2=(ct)^2-(x)^2$ ...where $t$ is the time between the events and $x$ is the distance between the events. When its ...
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Lorentz Invariance of the Euler-Lagrange equation for fields

Given an Lorentz invariant Lagrangian density $L$ of a Lorentz invariant scalar field $\phi$, How does one show that the following term in the Euler-Lagrange equation is invariant under Lorentz ...
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Charge is relativistic invariant. What does it mean? [duplicate]

Yesterday my teacher stated this statement that charge is relativistic invariant But he didn't explained this statement. So can anyone explain me this statement?
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Casimir operators for Poincare algebra

I have seen at various places the comment that the operator $P_\mu P^\mu$ is a Casimir operator of Lorentz algebra and thus it satisfies a on-shell condition like $P_\mu P^\mu=m^2$. Given the Poincare ...
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Variables of an SO(3)-invariant function (hamiltonian)

I am looking to reduce the dependence of a function, knowing that it satisfies some invariance constraints. Let me first formulate my question by explaining the 2-dimensional case. Imagine I have a ...
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Why does invariance commute with partial derivative?

This question applies more generally to actions, but I am going to ask it for a specific example. I am getting confused when considering the invariance of the superstring action under Weyl ...
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Why can't the constancy of the speed of light be deduced from classical physics? [closed]

I have read over a dozen questions about the speed of light -- "why it $c$ constant?", "why can't anything travel faster than light?", "how do we know this?" The responses are quite clear: The ...
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Why Pauli matrices are the same in any frame? [duplicate]

On page 157 of Schwartz's QFT book, He write that “$\sigma_i$ do not change under rotations”. If so, changes in $\psi$ and $B$ cancels, so we can get that $(\vec{\sigma} \cdot \vec B)\psi$ is ...
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How to show that scalar fields are translation invariant?

Classical scalar fields governed by Klein-Gordon equation, $$\left(-\frac{\partial^2}{\partial t^2}+\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}+m^2\...
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1answer
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Invariance under rotation of the Levi-Civita tensor

I'm trying to follow the answer to this post, that cites the identity $$ \epsilon_{i_{1} \ldots i_{n}} A_{j_{1}}^{i_{1}} \cdots A_{j_{n}}^{i_{n}}=\operatorname{det} A \epsilon_{j_{1} \ldots j_{n}} $$...
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What is the problem with a generalized kinetic term as $K^{\mu\nu}(x)\partial_\mu\phi\partial_\nu\phi$?

For field theory in flat spacetime, the most general kinetic term that I can think of for a field is $$K^{\mu\nu}(x)\partial_\mu\phi\partial_\nu\phi$$ where $K^{\mu\nu}(x)$ is an arbitrary second rank ...