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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Other ways to explain invariance? [closed]
I'm wondering if there are other ways to explain the invariance of the speed of light, besides the Lorentz/SR approach?
I heard Vladimir Ignatowski proved that the Lorentz transformation was the only ...
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Relation between $SL(2,R)$ and $U(1)$ symmetry
I have an action that I have proven to be invariant under an $SL(2,R)$ symmetry. But I actually want my action to be invariant under an $U(1)$ symmetry (because i know that for the system I am ...
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Physical meaning of the invariance of the dot product of $\vec{E}$ and $\vec{B}$
I recently learned about the fact that $\vec{E} \cdot \vec{B}$ is invariant under Lorentz transformations, which seems like a really nice and useful result. Is there a physical meaning similar to the ...
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Invariants from the covariant derivatives of a scalar field
I am reading Theoretical minimum: Special Relativity and Classical Field Theory where you construct a Lagrangian for the field by the argument that it would be invariant under the Lorentz ...
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Why is Noether's theorem not guaranteed by calculus?
The action of a system, say a scalar field is
$$ S = \int_{\mathcal{M}} {\rm d}^4 x ~ \mathcal{L}(\phi(x),\partial \phi(x)). $$
Now, if one does a variable transformation $x \to x'$, then
$$ S' = \...
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How one shows a law is invariant/variant under a transformation in general case?
Say we have a law of physics in a form of a differential equation:
$$
F(x,y,\dot{y},\dots) = 0.
$$
What is the general algorithm to show that this law is invariant under a certain transformation. The ...
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Invariant nature of mass and particle annihilation [closed]
Since mass is a Lorentz invariant, it can never change to preserve the vectorial nature of the four-momentum and the other four vectors. Thus the only interpretation of the energy-mass equation that I ...
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Confusion about Lorentz invariance of scalar product
I am a bit confused about the way that Lorentz invariance of the scalar product $A^\mu g_{\mu\nu}B^\nu$ is proved. Usually, the proof would go like this (see also e.g. this Physics SE question). The ...
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Why the Pauli-Lubansky and momentum operator build an irreducible representation of Poincarè group?
We know that particle states in QFT are identified with irreducible representation of Poincarè group, in particular they can be identified using Pauli-Lubansky and (squared) Momentum operator (wich ...
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Meaningful topological invariants and quantities for the description of 3D topological insulators
I'm currently trying to understand the classification of 3D topological insulators (like Bi2Se3). Most reviews dealing with this topic start with the introduction of the Quantum Hall effect since this ...
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Is there meaning in 'metric is invariant under transformation'?
In component (matrix) form of tensor, $I$ is invariant under Rotation and Minkowski metric is invariant under Lorentz transformation. Of course, scalar multiplication on $I$ or Minkowski metric does ...
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Forces that are invariant under Galilean spacetime rescaling $\mathbf x' = \lambda \mathbf x$, $t' = \lambda^2 t$
Consider a force of the form
$$
m \ddot{\mathbf x}(t) = -k\frac{\mathbf x(t) - \mathbf x_0}{|\mathbf x(t) - \mathbf x_0|^d}.
$$
For what values of $d$ is this force invariant under the Galilean ...
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Meaning of rotational invariance breaking and lifting of degeneracy
The $2p$ level of hydrogenic atoms are $3-$fold degenerate. That is $\psi_{210}$ and $\psi_{21\pm1}$ all have the same energy $E_{n=2}$. And the level $n=2$ as a whole is $4-$fold degenerate, since $...
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What is the correct predicted perception of a tennis ball moving at $c/2$ in Minkowski space?
Tennis ball $A$ sitting at $x=0$ and moving at velocity $v=0$ has a long thin rod attached to it of length $c/2$ meters pointed down the x axis. At $t=0$ tennis ball $B$ traveling at $v=c/2$ to the ...
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Invariance and covariance [duplicate]
What exactly do we mean when we say that something is covariant?
How is it different from being invariant?
Am i right if I say that the word invariant is used when we're talking about a physical ...
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Topological Invariants of Floquet Topological Insulators
I've been working through the following sources:
https://journals.aps.org/prb/abstract/10.1103/PhysRevB.96.155118
https://journals.aps.org/prb/abstract/10.1103/PhysRevB.96.195303
where they derive new ...
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From Wigner classification to field equations
Wigner classification of elementary particles tells us how many different particle types we should expect if we want their dynamic to be invariant under the Poincare group. The classification is done ...
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Is the state of a system invariant or not?
In a physics textbook by Eric Mazur there is the following phrase:
“The state of any object or system cannot depend on the motion of the observer.”
Here, the state is understood as some complete set ...
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How to generate or guess an invariant quantity?
What are the methods for guessing the invariant quantity:
$$x^2 - t^2$$
under Lorentz transformation
$$x'=\gamma(x-vt)$$
$$t'=\gamma\left(t - {vx}\right)$$
where c = 1.
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What is a simple example of Lorentz invariant equation? [closed]
The algebraic criteria for proving Lorentz invariance is that, replacing x with x' and t with t', where
$$x'=\gamma(x-vt)$$
$$t'=\gamma\left(t - \frac{vx}{c^2}\right)$$
should give back a certain &...
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Invariance over Galilean transformation [closed]
I want to prove that the Wave Equation is not invariant under Galilean Transformation. I'm having a little trouble with it but this is my attempt.
1. First of all, what does it mean by "not ...
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What is meant precisely, when a term in the Lagrangian is "chirally invariant"?
I am reading this paper, where eq. (2):
$$ m_0(\bar{\phi}_{-L}\phi_{+R}-\bar{\phi}_{-R}\phi_{+L}-\bar{\phi}_{+L}\phi_{-R}+\bar{\phi}_{+R}\phi_{-L}) \tag{2} $$
is said to be chirally invariant. Here, ...
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Is there a standard name for Robb's spacetime invariant equation?
I'm not sure exactly how to categorize Robb's treatment of the spacetime interval. But it seems like a gem of simplicity and insight.
The following illustrations are based on MTW Box 1.3. As drawn, ...
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Relation between pfaffians and Chern number
I have a 2D topological superconductor in symmetry class D, only particle-hole symmetry, nothing else. I can calculate the Chern number using
$$
C = \frac{1}{2\pi}\int_{BZ}\mathrm{d}^2{k}\ \mathcal{F}
...
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In Landau & Lifshitz's 'Theory of elasticity' why are scalars $u_{ik}^2$ and $u_{ii}^2$ said to be independent? They vary with the reference system
Scalars $u_{ik}^2$ and $u_{ii}^2$ depend on the reference system (are not invariants), why are they said to be independent? I have tried to rotate the reference system for a given tensor but $F$ ...
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Laws of physics invariant under proper orthochronous Lorentz Transformations - experimental fact or mathematically derived?
We know that the laws of physics are invariant under proper orthochronous Lorentz transformations.
How did we come to this knowledge? Is it an experimental fact that has not been violated, or can it ...
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Lorentz covariance vs invariance for $x_\mu p^\mu$
Is $x_\mu p^\mu$ Lorentz invariant and covariant?
I thought for a quantity to be Lorentz invariant, it should have the same value in every frame. However, unlike $p_\mu p^\mu = m^2$, $x_\mu p^\mu$ ...
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Charge, Parity, and Time are considered when talking about symmetries in physics. What about Rotational symmetry?
I've seen Charge, Parity, and Time symmetries talked about, but how come never rotational symmetry? E.g. if the entire universe was rotated 90 degrees, would any physical phenomenon behave differently?...
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Physical interpretation of invariant interval
I know that the invariant interval $I$ is the same in all reference frames. However, I don't know what is the physical meaning of $I$. Is it just a quantity for us to check our answers?
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Does the energy-momentum relation showcase the "magnitude", in the vectorial sense, of the energy?
The energy-momentum relation is:
$$E^2=(mc^2)^2+(pc)^2 \rightarrow E=\sqrt{(mc^2)^2+(pc)^2}$$
Which is obviously very similar to the magnitude of a vector: $|\textbf{v}|=\sqrt{v_1^2+v_2^2}$
This begs ...
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Operators that are invariant under rotation
Consider a system described by state $|\psi\rangle(x,y,z)$.
$\hat U_n(\theta)$ is an operator which rotates the wavefunction about an axis $n$ by an angle $\theta$ in positive direction.
$|\psi\rangle(...
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How does it make physical sense to define mass in terms of magnitude of the 4-momentum?
One can show that for a point particle of mass $m$, in Special Relativity its energy in terms of the velocity is given by $$E=\frac{mc^2}{\sqrt{1-\frac{v^2}{c^2}}}. \tag 1$$
A remarkable feature of ...
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Lack of Rotational Invariance in Relativistic Quantum Mechanics
In 'Lectures of Sidney Coleman on Quantum Field Theory' the first paragraph of the first chapter briefly discusses the lack of rotational invariance in relativistic quantum mechanics. I will quote ...
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Why is the action an adiabatic invariant in a unidimensional oscillator?
I'm reading Rax's "Méchanique Analytique" but I can't understand a particular step.
We consider a unidimensional oscillator system with a potential that depends on a parameter $\lambda(t)$ ...
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Proper time while going up in a gravitational field
Feynman, in his book 'Surely You're Joking, Mr. Feynman' refers to a problem he once gave one of Einstein's assistants at Princeton. The problem goes something like this:
You blast off in a rocket ...
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Calculus of Variations, Noether's Theorem, need help understanding and solving this problem
I have been given the following problem:
Investigate whether the functional $I=\int_{t_1}^{t_2}t\dot{x}^2dt$ is invariant under the transformation $\bar{t}=t+\epsilon$ and $\bar{x}=x$, with $\epsilon$...
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Choice of scalars in Einstein-Hilbert action
I read several questions, but I can't find the answer to these two
How many non-trivial independent scalar quantities can we extract from a curved spacetime and why just $R$ appears in the action?
...
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Lorentz transformations in Minkowski space including invariants
Let's assume the reference frame of a still observer with the 2 axes: the x-axis and time axis t. Let's say there's another observer within that frame moving with a constant velocity v with respect to ...
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Why fields carrying spin cannot obtain non-zero vacuum expectation value in a Lorentz invariant theory? [duplicate]
Why fields carrying spin cannot obtain non-zero vacuum expectation value in a Lorentz invariant theory?
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How to show $SU(3)$ symmetry of the following hamiltonian?
I have a hamiltonian of the form:
$H = \sum_i (\hat{U^+_i}\hat{U^-_{i+1}} + \hat{U^-_i}\hat{U^+_{i+1}}+\hat{V^+_i}\hat{V^-_{i+1}}+\hat{V^-_i}\hat{V^+_{i+1}}+\hat{T^+_i}\hat{T^-_{i+1}}+\hat{T^-_i}\hat{...
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Srednicki's QFT: Why $\langle p|\phi(0)|0\rangle$ in the interacting theory is Lorentz invariant?
I am reading Srednicki's QFT and I have met a problem. In its section 5, (5.18) , after deducing the LSZ formula, in order to check whether his supposition "that the creation operators of free ...
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Are Pauli matrices invariant tensors in the representation of $\frac12 \otimes \frac12 \otimes 1$?
If we raise the index of the Pauli matrices with Levi-Civita symbol $\epsilon$ we obtain the 2-index spinors $(\sigma_i)^{AB} = (\sigma_i)^A{}_C \ \epsilon^{CB}$. The textbook (Ref. 1) argued that ...
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Is the Dirac Lagrangian a 4-vector?
I was under the impression that the Lagrangian outputs the energy, which is a real-valued scalar.
However, the Dirac Lagrangian seems to read as a 4-vector:
$$
L= \overline{\psi} (i\hbar \gamma^\mu \...
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What is the theoretical value of this phase space invariant?
So I wanted know how to theoretically calculate this phase space invariant (equation $3.31a$ )$R$ in our universe (FLRW metric) during the cosmological nucleosysthesis:
$$R = \int_{p} \frac{\mathcal{...
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Why does contracting a term with a tensor means a portion of this term is a tensor?
I am looking at a problem in Guidry's Modern General Relativity, and the solution contains the following two sentences:
In the scalar product expression $A\cdot B = g_{\mu \nu}A^{\mu} B^{\nu}$, the ...
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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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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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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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