# 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 ...
1 vote
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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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1 vote
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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 $p_\mu p^\mu = m^2$, $x_\mu p^\mu$ ...
1 vote
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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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### 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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### 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 ...
1 vote
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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 ...
217 views

### 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?