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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26 views

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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25 views

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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2answers
205 views

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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1answer
64 views

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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1answer
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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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3answers
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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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1answer
117 views

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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2answers
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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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2answers
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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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2answers
235 views

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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1answer
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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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1answer
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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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3answers
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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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What is the difference between 'Rest Mass', 'Centre of Mass' and 'Invariant Mass' 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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1answer
68 views

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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45 views

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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2answers
36 views

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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1answer
93 views

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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0answers
31 views

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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2answers
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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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1answer
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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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67 views

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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5answers
2k views

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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3answers
547 views

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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2answers
135 views

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
124 views

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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2answers
82 views

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 ...
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1answer
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What is actually meant when it is said Scalar is invariant?

As far as i know a quantity is called invariant if it satisfies some specific transformations. Now,Suppose a body is moving with velocity $\vec{v}$ as measured from the lab frame.Its non-relativistic ...
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1answer
36 views

Which quantities are Lorentz-Invariable and which are not? [duplicate]

The only physical constant I know for sure is Lorentz-Invariant is electric charge. I am curious to know if there are others even if it's not possible to make an exhaustive list.
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Is Lorentz-Invariant opposite to Lorentz-covariant? [duplicate]

I am having trouble understanding the meaning of these terms. Is it possible to be both Lorentz-Invariant and Lorentz-Covariant at the same time?
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1answer
49 views

What quantities are Lorenz-Invariant? [closed]

I understand that quantities in physics can either be Lorenz-Invariant (eg charge) or Lorenz-Covariant (eg length). Is it possible to obtain an exhaustive list of which quantities are which?
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1answer
45 views

Poincaré Group element of pure spacetime translation

If we make a spacetime translation of the coordinates of a event $ x^{\mu}$ such that $x' ^{\mu} = x ^{\mu} + a^{\mu}$, the element $\eta _{\mu \nu} x'^\mu x' ^\nu $. Must be invariant : \begin{...
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1answer
28 views

Adiabatic Invariant for nonoscillatory system

For oscillatory system (e.g. quantum harmonic oscillation with slowly changing effective spring constant), it is common to define the adiabatic invariant to be $$I=\frac{H(t)}{\omega(t)}$$ where $H$ ...
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Manifestly Lorentz invariant nature of the equation $\phi(x)=\int\frac{d^3k}{(2\pi)^3\sqrt{2\omega_k}}(a_k e^{-ikx}+a^{\dagger}_ke^{ikx})$ [duplicate]

Quantum field for Klein-Gordon Equation is defined as $$\phi(x)=\int\frac{d^3k}{(2\pi)^3\sqrt{2\omega_k}}(a_k e^{-ikx}+a^{\dagger}_ke^{ikx})$$ Please forgive me for my slopiness here regarding the ...
4
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1answer
883 views

Which quantity is mass (tensor, vector or scalar)? [closed]

Mass and spin are fundamental characteristics of particle. Those quantities are eigenvalues of the Casimir operators of the Poincaré group. My book then writes that $$ p^\mu p_\mu = m_0^2, $$ where $...
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1answer
102 views

Find the Scalar Invariant under a Lorentz-transformation [closed]

[Question] Given the components of two vector fields, $u^\alpha$, $v^\beta$, show that $u^\alpha v^\alpha = u^0 v^0 + u^1 v^1 + u^2 v^2 + u^3 v^3$ is not a scalar invariant under a Lorentz-...
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1answer
90 views

How are the Euler-Lagrange Equations any more coordinate-invariant than Newton's?

In my experience it is often said that the Lagrangian formulation of mechanics can be much much more convenient because the form of the (E-L) equations remains the same whatever coordinates we choose, ...
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0answers
56 views

Curvature and Symmetries of spacetime

Is there any relation between symmetries of spacetime and the curvature invariants? For example is spherical symmetric spacetimes, necessarily have positive curvature? Could we define any spherical ...
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1answer
128 views

What is the difference between invariance and covariance? [duplicate]

In relativistic physics, paricularly in General Relativity and Quantum Field Theory, we often find the use of the two terms 'invariance' and 'covariance'. But I couldn't find any mention of the ...
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1answer
137 views

Chern-Simons action as a topological invariant

It is stated that the Chern-Simons action is a topological invariant that is proportional to the Chern-Simons form. But the latter is just a conformal invariant. How do we reconcile these views? Both ...
0
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2answers
41 views

What are the analogues of momentum, inertia and angular momentum for energy?

If Energy and mass are the same thing, then is it logical to look for analogous (duals) of properties of one in another? or there is there any conceptual framework that such questions make any sense? ...