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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.

Invariance is the property of remaining unchanged regardless of changes in the conditions of measurement. Without invariance principles, there would be no laws of physics. From the invariance view we naturally arrive at a consideration of symmetries and structures. It is often claimed that there is a strong connection between invariance and reality, established by symmetries. The invariance view seems to render frame-invariant properties real, while frame-specific properties are illusory.

Importance: Invariants are important in modern theoretical physics, and many theories are expressed in terms of their symmetries and invariants.

Covariance and contravariance generalize the mathematical properties of invariance in tensor mathematics, and are frequently used in electromagnetism, special relativity, and general relativity.

References:

http://philsci-archive.pitt.edu/3643/1/Invariance%2CSymmetries...pdf

http://www.sjsu.edu/faculty/watkins/physicsinvar.htm

https://en.wikipedia.org/wiki/Invariant_(physics)

353 questions
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To which extent is general relativity a gauge theory?

In quantum mechanics, we know that a change of frame -- a gauge transform -- leaves the probability of an outcome measurement invariant (well, the square modulus of the wave-function, i.e. the probability), because it is just a multiplication by a…
FraSchelle
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Fundamental invariants of the electromagnetic field

It is a standard exercise in relativistic electrodynamics to show that the electromagnetic field tensor $F_{\mu\nu}$, whose components equal the electric $E^i=cF^{i0}$ and magnetic $B_i=-\frac12\epsilon_{ijk}F^{jk}$ fields in the taken frame of…
Emilio Pisanty
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3 answers

Definition of Casimir operator and its properties

I'm not sure which is the exact definition of a Casimir operator. In some texts it is defined as the product of generators of the form: $$X^2=\sum X_iX^i$$ But in other parts it is defined as an operator that conmutes with every generator of the Lie…
jinawee
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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' = \int_{\mathcal{M}'} {\rm d}^4 x' ~…
Faber Bosch
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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 synonyms? "Invariant" on the other hand refers to…
Josh
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Is the "number of photons" of a system a Lorentz invariant?

I'm wondering whether the number of photons of a system is a Lorentz invariant. Google returns a paper that seems to indicate that yes it's invariant at least when the system is a superconducting walls rectangular cavity. However I was told in the…
untreated_paramediensis_karnik
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Invariance, covariance and symmetry

Though often heard, often read, often felt being overused, I wonder what are the precise definitions of invariance and covariance. Could you please give me an example from field theory?
Hamurabi
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Proving invariance of $ds^2$ from the invariance of the speed of light

I've started today the book of Landau and Lifshitz Vol.2: The Classical Theory of Fields $\S 2$. They start from the invariance of the speed of light, express it as the fact that $$c^2(\Delta t)^2-(\Delta x)^2-(\Delta y)^2-(\Delta z)^2=0$$ is…
user35543
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Invariance of action $\Rightarrow$ covariance of field equations?

Invariance of action $\Rightarrow$ covariance of field equations? Is this statement true? I have only seen examples of this, like the invariance of Electromagnetic action under Lorentz transformations. How could we prove it? The action is a scalar,…
SuperCiocia
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Rank of the Poincare group

There are two Casimirs of the Poincare group: $$ C_1 = P^\mu P_\mu, \quad C_2 = W^\mu W_\mu $$ with the Pauli-Lubanski vector $W_\mu$. This implies the Poincare group has rank 2. Is there a way to show that there really are no other Casimir…
Neuneck
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3 answers

Why are the metric and the Levi-Civita tensor the only invariant tensors?

The only numerical tensors that are invariant under some relevant symmetry group (the Euclidean group in Newtonian mechanics, the Poincare group in special relativity, and the diffeomorphism group in general relativity) are the metric $g_{\mu \nu}$,…
tparker
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2 answers

Difference between symmetry and invariance

I'm wondering what's the real difference between symmetry and invariance in Physics? I believe that sometimes the two words are given the same meaning and some other times they are used in a different way. To me a symmetry is more of a physical…
user78618
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Does the Kosterlitz–Thouless transition connect phases with different topological quantum numbers?

The Kosterlitz-Thouless transition is often described as a "topological phase transition." I understand why it isn't a conventional Landau-symmetry-breaking phase transition: there is no local symmetry-breaking order parameter on either side of the…
tparker
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What exactly does it mean for a scalar function to be Lorentz invariant?

If I have a function $\ f(x)$, what does it mean for it to be Lorentz invariant? I believe it is that $\ f( \Lambda^{-1}x ) = f(x)$, but I think I'm missing something here. Furthermore, if $g(x,y)$ is Lorentz invariant, does this means that…
QuantumEyedea
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Why is mass an invariant in Special Relativity?

I have read here that mass is an invariant and that it is the momentum that approaches infinity when your speed approaches the speed of light. That is why infinite energy is required to accelerate an object to the speed of light. But, why doesn't…
PNS
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