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Questions tagged [invariants]

The tag has no usage guidance.

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How is mass related to the Poincare group?

I have studied about Poincaré group and some QFT, read that the Casimir elements are $p^{u}p_{u} = m^2$ and Pauli-Lubanski vector (or pseudovector). How does the mass come into picture and spin come ...
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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
76 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
124 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
52 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 ...
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3answers
377 views

How can zero point energy vacuum be Lorentz invariant?

What distribution of electromagnetic radiation is Lorentz invariant? How can radiation look the same regardless of inertial frame? According to Marshall and Boyer a cubic distribution like $ \rho(\...
3
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1answer
97 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
69 views

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
52 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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1answer
76 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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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. ...
2
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1answer
91 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
130 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 ...
0
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1answer
70 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 ...
2
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1answer
83 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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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 ...
2
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1answer
153 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 ...
2
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2answers
146 views

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
67 views

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
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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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Why Casimir in super-Poincare is zero in massless case?

Trying to find the Casimir's of super Poincaré algebra, Calculating in the massless case: $P_{a} = (E, 0, 0, E).$ The Casimir: $C_{ab} = B{a}P{_b} - B_{b}P_{a} = Square[BP] - Square[B]Square[P]$ is ...
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1answer
145 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 ...
0
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1answer
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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 ...
3
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1answer
126 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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0answers
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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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0answers
36 views

Null Electromagnetic Field=Field with no sources?

I get that a null electromagnetic field is a field for which the fundamental invariants vanish. Every example I come across is that of some type of pure radiation. While I haven't found it ...
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0answers
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Do coordinates that make the Minkowski space interval invariant necessarily transform as the Lorentz transformations? [duplicate]

Let's say that some coordinates in a Minkowski space transform in some way which is unknown. However, it can be proved that the Minkowski space interval is invariant under whatever transformations ...
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0answers
178 views

Why are actions Lorentz Invariant?

I understand what actions are but how do we know that they are manifestly Lorentz invariant? And also, is there a mathematical rule there for an object to be Lorentz invariant?
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3answers
80 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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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
277 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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0answers
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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 ...
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3answers
670 views

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 ...
2
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1answer
160 views

importance of invariant tensors

while studying representations of SL(2,C), for raising and lowering indices of spinors invariant tensor $\epsilon$ was constructed analogous to $\eta$ in SO(1,3).What is the importance of invariant ...
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3answers
302 views

The implications of Einstein's first law

I'm struggling with the physical meaning/consequence of Einstein's first postulate of Special Relativity, which states that all physical laws are the same (invariant) in all inertial frames. Any ...
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2answers
453 views

Has the energy-momentum invariant any meaning?

For a single particle system we have: $$E^2 - (pc)^2 = (mc^2)^2.$$ In my lecture notes it has also been stated that for a system of several particles: $$\left(\sum E\right)^2 - \left(\sum p\right)^2c^...
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2answers
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Why do we have $\phi'(x')=\phi(x)$ for a field satisfying Klein-Gordon equation?

I would like to know why we have $\phi'(x')=\phi(x)$ for a field satisfying Klein-Gordon equation. Is it an assumption or can it be proved? The $'$ means a Lorentz transformation: $\phi'$ is the ...
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1answer
897 views

A physical considerations on Lorentz invariant $E^2-c^2B^2$

I have see this explanation of Lorentz invariant $E^2-c^2B^2$: Precisely, what does it refer $\widetilde{B}$ or $\widetilde{E}$? After, on the site Why is this invariant in Relativity: $E^2−c^2B^2$? ...
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0answers
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rotational invariance

I have been treating this Hamiltonian: $$H=v\vec{p}\vec{\sigma}+\vec{A}\vec{\sigma}$$ where $\vec{\sigma}=(\sigma_x , \sigma_y)$. It is relevant for 2D graphene quantum dots with some vector ...
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2answers
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Why $x^2_0-x^2_1-x^2_2-x^2_3$ is invariant under $O(1,3)$?

What exactly means that a certain mathematical statement is invariant under a group? For Ex:$$O(1,3)$$ for $$x^2_0-x^2_1-x^2_2-x^2_3$$ and how do you check it?
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1answer
158 views

Srednicki QFT ch34: invariants of the Lorentz group

In chapter 34 of his Quantum Field theory handbook, Srednicki discusses invariants of the Lorentz group and how they appear in the decomposition in irreducible representations of Lorentz tensors. As ...
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2answers
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Prove that $\mathbf{E}^2-\mathbf{B}^2$ and $\mathbf{E}\cdot\mathbf{B}$ are the only two independent Lorentz invariant quantities [duplicate]

How to prove that $\mathbf{E}^2-\mathbf{B}^2$ and $\mathbf{E}\cdot\mathbf{B}$ are the only two independent Lorentz invariant quantities that are constructed by $\mathbf{E}$ and $\mathbf{B}$? It's ...
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1answer
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How to identify cosntants of motion/ which constants of motion are independent of mass?

I was asked to identify a constant of motion which does not depend on the mass of the object whilst in contact with the surface. I found the equation of motion of the material point but I don't know ...
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3answers
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Definition of the Spacetime Interval

The spacetime interval is defined as follows: $$\Delta s^2 = -(c\Delta t)^2 + \Delta x^2 + \Delta y^2 + \Delta z^2$$ or in tensor notation: $$\Delta s^2 = \eta_{\mu\nu} \Delta x^\mu \Delta x^\nu$$ ...
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Defining invariant spacetime interval

So, my textbook goes about defining the invariant spacetime in the following way: Consider two frames of references, S and S', with a relative speed to each other, coinciding at t=t'=0. At t=0, a ...
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3answers
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Strange result from braking energy in different reference frames. Spot the error [closed]

Assume a truck of mass 1 ton braking from 60mph to 0 on a road surface. From the reference frame of the road, the following energy is transferred to the brakes: 1 ton * 60mph^2 / 2 = 1800 ton mph. ...
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1answer
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Proving an invariant relationship [closed]

Two particles, moving at relativistic speeds in the x direction, are observed to have energies E1 and E2, and momenta p1 and p2 in frame A. Frame B moves at relativistic speed v relative to frame A, ...
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1answer
679 views

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 ...