# Questions tagged [invariants]

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106 questions
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### 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 invariency of compton scattering, you can prove that for spin 1 field there ...
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### Eigenvalues of quadratic Casimirs of simple Lie groups

I want to find a generic formula for calculating eigenvalue of quadratic casimirs of Lie groups, in terms of Dynkin labels. For a simple example if we take $SU(2)$, with $[R]$ indicating the highest ...
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### Three questions and explanations for the Lorentz invariant $E^2-c^2B^2$

It is demonstrated that the square trace of the electromagnetic tensor is nothing and it is valid: $$\mathrm{Tr}\,{F}^2_{\mu\nu}=\frac{2}{c^{2}}(E^2-c^2B^2).$$ Proof: $F_{\mu\nu}=-F_{\nu\mu}$, hence ...
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### Given a metric of a torus can we measure it's thickness?

What I mean is, if you have a metric of a torus $T_2$, and you want to distinguish between say very thin stringy tori and very thick tori with the same surface area, is there a nice formula for this ...
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### Invariant mass in special relativity [closed]

I'm following a special relativity course and I'm trying to understand how the invariant mass works. In particular I don't get how the following passages work. We have a collision between two ...
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### Definitions and usage of Covariant, Form-invariant & Invariant?

Just wondering about the definitions and usage of these three terms. To my understanding so far, "covariant" and "form-invariant" are used when referring to physical laws, and these words are ...
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### Issue showing that the phase of a harmonic wave is invariant under a Galilean transform

The phase $Φ$ of wave is defined as $kx-wt$. It should be the case that all observers moving relative to each other in the non relativistic case will agree on this. So given the transforms $x'=x-vt$ ...
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### Invariance of length [closed]

Invariance of interval in Minkowski space under coordinate transformation was proved by the postulates of special relativity. (https://physics.stackexchange.com/a/453536/213658 .see this answer) Is ...
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### Newton's theory of gravity is covariant under Galilean transformations

We know from classical mechanics that the gravitational field equation for the scalar potential takes the form $$\nabla^2\phi=4\pi \rho,$$ where $\rho$ is mass density (which, can depend on time and ...
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### How length is an invariant in Euclidean space?

The special theory of relativity shows that intervals are invariant under Lorentz transform in the Minkowski space -time. But how can we prove (any postulates or theory) that the length is an ...
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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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### What is the difference between a physical constant, a scalar, an invariant, and a conserved quantity?

I don't really know how to properly articulate this question. This question popped into my mind when pondering why the fact that a physical constant like the speed of light doesn't have an associated ...
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### Why do we differentiate a 4 vector with respect to proper time to obtain 4-velocity?

The coordinates of an event in spacetime are given by the 4-vector $(ct, \mathbf{r})$, where $\mathbf{r}$ is the spacial coordinates of the event. This 4-vector can be seen as 4-displacement of a ...
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### Invariance of the relativistic interval

Imagine we have two events, $E_1, E_2$ in the coordinate systems $K, K'$ (with coordinates $(x,y,z,t),\ (x',y',z',t')$), whilst $K'$ ist moving with the speed $\vec v$ in regard to $K$. From the ...
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### Converting an invariant matrix to a non-invariant tensor

I'm working on the following problem: In 4-dimensional notations, given a transformation matrix Calculate the matrices $\Lambda_{\mu\nu}$, $\Lambda_\mu^\nu$ and $\Lambda^{\mu\nu}$ The ...
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### Special relativity: I arrive at a contradiction regarding the Lorentz invariance of certain quantity

I want to show the Lorentz invariance of $d^3 p/E$ (Eq. 8.11 of Mandl-Shaw), where $E$ is the relativistic energy. Peskin-Schroeder gives sort-of, a proof in section 2.3 which I am convinced of. But ...
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### How does one prove the channel independent inequality satisfied by the product of the three Mandelstam variables?

How does one prove the following equation (67.5) from the BLP Quantum Electrodynamics book? The q's are the 4 momenta, and h is the sum of all four masses. Two q's written after one another in the ...
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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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### 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'^...