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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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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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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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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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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612 views

Finding Casimir operators for the Poincare group $ISO(1,2)$

I was asked to write the generators for translations and Lorentz-transforms in 1+2 dimensions and then to find the Casimir operators. For the generators I can take the same ones as in 1+3 case $$P_\...
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1answer
63 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
133 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 ...
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114 views

Scalar versus invariant in Newtonian mechanics

I looking up coriolis transport theorem for rotating refrence frames and while reading through this derivation he wrote: In Newtonian mechanics, scalar quantities must be invariant for any given ...
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145 views

Explicit Quadratic Casimir for $sp(2N)$

We know that $so(3)$ has the explicit quadratic Casimir $$L^2=\sum L_{i}^2.$$ Are there analogs to this in other simple lie algebras? I know that for a simple lie algebra I can always use the ...
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3answers
101 views

Interval Preserving in Minkowski Space

The squared line element in any spacetime is given as $$ds^{2}=g_{ab}dx^{a}dx^{b}.$$ The use of tensors helps us to infer that the line element in some other frame would be $$ds'^{2}=g'_{ab}dx'^{a}dx'^...
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Use of step function to show that total charge is Lorentz invariant if the four-gradient of current density is zero

I am reading the chapter on Special Relativity in Steven Weinberg's 'Gravitation and Cosmology'. It is stated in the book that total charge can be written as $Q=\int d^4xJ^{\alpha}(x)\partial_{\alpha}\...
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306 views

rotational invariance

I have been treating this Hamiltonian: $$H=v\vec{p}\vec{\sigma}+\vec{A}\vec{\sigma}$$ where $\vec{\sigma}=(\sigma_x , \sigma_y)$. It is relevant for 2D graphene quantum dots with some vector ...
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428 views

SU(3) adjoint representation's invariant tensors

Considering a complex scalar field $\varphi^a$ that transform in the adjoint representation (8) of SU(3). A quartic interaction term SU(3) invariant is $$\lambda C^{abcd}\varphi^{\dagger a} \varphi^...
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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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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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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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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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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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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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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
49 views

time invariance for “Translations” versus “Galilean transformations”

Why would the time coordinate (t) be NOT invariant under translations, but invariant under Galilean transformations? I thought it should be invariant under both Here is what I'm tying to understand:
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1answer
260 views

Gravitons and self-interaction

In the book quantum field theory and standard model by Schwartz, there is a problem 9.4 that says by considering lorentz invariance of Compton scattering, you can prove that for spin 1 massless field ...
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A Scalar Function Tranformation — Question on Notation in 't Hooft Document

I started reading a document by Gerard 't Hooft which can be found here. Right at the start I am puzzled by a simple expression. It is equation 2.2 showing how a scalar function transforms. I ...
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2answers
171 views

Character expansion and Casimir

Is there a simple way to extract the quadratic Casimir of a representation from the character? I keep hearing things such as "Chern characters have an expansion that goes like" $$\chi(r) = dim(r) ...
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Building the Lagrangian of electromagnetism from the Lorentz invariant?

The definition of the relativistic Action is $$ S=\int_a^b ds $$ The Lorentz invariant of electromagnetism is $$ s^2=\frac{1}{c^2}||\mathbf{E}||^2-||\mathbf{B}||^2-2i\frac{1}{c}(\mathbf{B}\cdot \...