# 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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### 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 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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### 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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### 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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### 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 ...
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 $$\int \frac{d^4 k}{(2\pi)^4}\delta(\omega^... 0answers 33 views ### Show invariance of the inner product of 4-velocities in different frames In the lab frame, particle B moves to the right with speed u, and particle C moves to the left with speed v. In the frame of C, particle B is seen to move to the right with speed w, ... 0answers 44 views ### Reparametrization of einbein action I would like to show that the following action$$ \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)} $$is ... 1answer 93 views ### Lorentz Invariance of the Euler-Lagrange equation for fields Given an Lorentz invariant Lagrangian density L of a Lorentz invariant scalar field \phi, How does one show that the following term in the Euler-Lagrange equation is invariant under Lorentz ... 1answer 90 views ### How are the Euler-Lagrange Equations any more coordinate-invariant than Newton's? In my experience it is often said that the Lagrangian formulation of mechanics can be much much more convenient because the form of the (E-L) equations remains the same whatever coordinates we choose, ... 0answers 56 views ### Curvature and Symmetries of spacetime Is there any relation between symmetries of spacetime and the curvature invariants? For example is spherical symmetric spacetimes, necessarily have positive curvature? Could we define any spherical ... 1answer 49 views ### time invariance for “Translations” versus “Galilean transformations” Why would the time coordinate (t) be NOT invariant under translations, but invariant under Galilean transformations? I thought it should be invariant under both Here is what I'm tying to understand: 1answer 260 views ### Gravitons and self-interaction In the book quantum field theory and standard model by Schwartz, there is a problem 9.4 that says by considering lorentz invariance of Compton scattering, you can prove that for spin 1 massless field ... 0answers 39 views ### A Scalar Function Tranformation — Question on Notation in 't Hooft Document I started reading a document by Gerard 't Hooft which can be found here. Right at the start I am puzzled by a simple expression. It is equation 2.2 showing how a scalar function transforms. I ... 2answers 171 views ### Character expansion and Casimir Is there a simple way to extract the quadratic Casimir of a representation from the character? I keep hearing things such as "Chern characters have an expansion that goes like"$$\chi(r) = dim(r) ...
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 \...