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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Why is it necessary that different observers agree on the value of the spacetime interval $ds^2$?

What's the physical reason that all (inertial) observers agree on the value of the spacetime interval $$ds^2 = (c dt)^2 - dx^2 - dy^2 -dz^2 \, ?$$ What would be the physical implications if different ...
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How could 'rest mass' and 'invariant mass' be the same?

The terms rest mass and invariant mass are often interchanged, however i cannot reconcile these concepts: Consider a photon ...
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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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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 \...
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2answers
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Why are certain quantities so fundamental to physics? [closed]

Apart from having a qualitative description of quantities such as momentum, work and energy, why are these quantities considered so fundamental? What is the reason to define them in the first place? ...
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1answer
280 views

Why helicity for massless particles is Lorentz invariant?

By definition helicity is projection of spin onto the 3 momentum. $$h={\bf J} \cdot {\mathbf{P }} $$ where ${\mathbf{P }}=(P_1,P_2,P_3)$ is the momentum operator and ${\mathbf{J }}=(J_1,J_2,J_3)$ ...
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1answer
114 views

General definition of symmetry in physics? [duplicate]

I've looked at a number of questions on what symmetries are in physics, such as this one, this one and this one. However, I found the questions and answers to be not completely satisfying because they ...
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1answer
217 views

Is Hamiltonian a scalar or tensor in Quantum Mechanics?

According to Wikipeida, a scalar operator is invariant under rotations, and the Hamiltonian satisfies this definition. But at the same time, a Hamiltonian can be written as a matrix, which means it is ...
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1answer
87 views

Antisymmetric matrices in effective field theory

I'm trying to construct a nonlinear $d$-dimensional version E&M as an effective field theory. Let $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$ be the field strength. The most general action ...
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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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2answers
47 views

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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2answers
90 views

Why do we need invariants to represent real life quantities?

Often it is said that one of the most useful properties of eigenvalues of a matrix is that they are invariant under change of basis. This in turn is said to be useful in physics because real, physical ...
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1answer
51 views

How to construct invariant forms under the effect of an arbitrary group?

First I would like to mention that I do not know that should I post this question here or in the math community, but since my background is in physics and this kind of question is usually asked by ...
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1answer
110 views

An invariant for transformations of Lorentz

Exist a physical demonstration why $$E^2- p^2c^2 =m^2c^4=E'^2- p'^2c^2 $$ is an invariant for transformations of Lorentz? N.B.: $m$ is mass; $E$ is the energy and $p$ is momentum in the frame $\...
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1answer
84 views

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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1answer
118 views

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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1answer
48 views

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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1answer
535 views

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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1answer
375 views

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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4answers
302 views

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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4answers
366 views

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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1answer
62 views

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 matrix $\...
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578 views

How can zero point energy vacuum be Lorentz invariant?

What distributions of electromagnetic oscillating fields are Lorentz invariant? How can oscillating electromagnetic fields look the same regardless of inertial frame? According to Marshall and Boyer ...
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1answer
123 views

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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1answer
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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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1answer
90 views

The Lorentz-invariant particle spectrum

Before the question, I need to mention some necessary definitions. The rapidity is defined as: $$y=\frac{1}{2}\ln\frac{E+p_z}{E-p_z}=\frac{1}{2}\ln\frac{1+v_z}{1-v_z}=\tanh^{-1}(v_z)$$ where $v_z=...
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3answers
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Why is the scalar product of two four-vectors Lorentz-invariant?

Why is the scalar product of two four-vectors Lorentz-invariant? For instance, given two four-vector $A^\mu$ and $B^\mu$, so their scalar product is $A\cdot B=A^\mu B_\mu=A^\mu g_{\mu\nu}B^{\nu}$. ...
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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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1answer
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Can GR be reformulated in terms of invariant observables?

Question So recently I was thinking about this: How many scalars are available in $4$ dimensions in General Relativity (without being redundant)? For example, with metric we can construct the ...
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1answer
414 views

Physical meaning of the Casimir operators of Poincarè algebra

If one considers the algebra $su(2)$, it is well known that the Casimir Operator is $$ C=L_1^2+L_2^2+L_3^2. $$ It corresponds to the total angular momentum and correctly is a conserved quantity. ...
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1answer
104 views

Value of the invariant $R_{\mu \nu}F^{\mu \nu}$

Is there a simple way to find the value of $R_{\mu \nu}F^{\mu \nu}$ (where $R_{\mu \nu}$ is the Ricci tensor and $F^{\mu \nu}$ is the electromagnetic tensor), knowing that it is an invariant? ...
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1answer
283 views

Can anyone provide a simple, inuitive explanation for Noether's Theorem? [duplicate]

I recently came across this theorem for the first time. As I understand it, what she showed was that conservation 'laws' are often simply an artifact of symmetry or invariance. For example, the ...
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1answer
111 views

How is Einstein's postulate about the invariance of the laws of physics justified? [duplicate]

According to one of Einstein's postulates related to special relativity, > "the laws of physics remain invariant in their form and nature in all inertial frames". But global inertial frames don't ...
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1answer
110 views

Restrictions on the form of a scalar-valued function imposed by Lorentz invariance

Let $f(p,q)$ be a smooth Lorentz-invariant function of 4-vectors $p$ and $q$. Should $f$ necessarily be of the form $f(p,q) = g(p^2, q^2, p_\mu q^\mu)$, where $g(x,y,z)$ is some scalar-valued ...
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0answers
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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3answers
471 views

Gauge invariant scalar which is not Lorentz-invariant

I'm looking for examples of the following descriptions: A gauge invariant scalar which is not Lorentz-invariant A Lorentz covariant scalar For 1. I was thinking about the scalar potential $A$ (for ...
2
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1answer
466 views

What is the physical meaning of the third invariant of the strain deviatoric?

In continuum mechanics of materials with zero volumetric change, the material condition can be expressed by the strain deviatoric tensor instead of the strain tensor itself. To express the plasticity ...
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2answers
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Are there Galilean scalars?

In special relativity there are scalar quantities which are invariant under any Lorentz transformation, called Lorentz scalars. For example, the magnitude of the four-velocity is a Lorentz scalar. If ...
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2answers
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Formal Term for an invariant constant to all observers

I was thinking of the speed of light and realized I don't know how to quickly name the concept of "physical quantity that is measured to be the same in all reference frames". Are there examples of ...
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1answer
146 views

Using the energy-momentum invariant for a decay process

For a decay process in which particle A ----> particle B + a photon in which particle A has mass $m_A$, particle of mass $m_B$ and energy and momentum are conserved. Show that in the frame in ...
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1answer
231 views

Quantities invariant by Lorentz transform

If I rotate a triangle lying on the two-dimensional plane both the lengths of the sides and the angles formed are invariant (ie, they are the same before and after the rotation). In a 2D Lorentz ...
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1answer
123 views

Why should an action in SR be a lorentz scalar? [duplicate]

I have taken it granted that an action in the special relativity must be a lorentz scalar. However is there a fundamental reason for this requirement? I cannot think of a plausible reason for this ...
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1answer
446 views

What exactly is an invariant quantity?

I have a bit of confusion regarding an invariant quantity. It is something which doesn't change on switching from one inertial frame to other like $\Delta$$\mu$$J$$\mu$ is an invariant. I read ...
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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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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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0answers
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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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What Lorentz covariance has to do with Lorentz invariance? [duplicate]

Does saying that the Dirac equation is invariant under Lorentz transformations is the same as saying that it is Lorentz covariant?
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2answers
452 views

Distance formula in Euclidean space vs. Spacetime Interval - why is one Pythagorean and one not?

I appreciate if this question has been posited before and easily findable by Google searching, but as of yet I haven't found anything to answer this. I'm sure I'm making an incorrect assumption in the ...
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Why is the Ricci scalar the only independent scalar constructed from products of the metric and its first and second derivatives? [duplicate]

In Sean Carroll's book, last paragraph of page 160, this statement is found: "The Riemann tensor is of course made from the second derivatives of the metric, and we argued earlier that the only ...