All Questions
Tagged with galilean-relativity group-theory
24 questions
8
votes
1
answer
119
views
Analogue of Coleman-Mandula theorem for non-relativistic quantum field theory?
For relativistic quantum field theories, the Coleman-Mandula theorem places very strong restrictions on the possible symmetry groups $G$ of the aforementioned quantum field theory, forcing it to be a ...
- 1,857
2
votes
1
answer
57
views
Are projective representiations of a Lie group a representation of the semi-direct product of the group with $U(1)$ if the norm is preserved?
Let's say we have a function $f(x_{\mu},t)$ that transforms under the action of an $N$-parameter group $G(a_{\nu})$. Then a projective representation of $G(a_\nu)$ in the $f(x_\mu,t)$ basis would ...
- 372
4
votes
0
answers
77
views
Using Galilean covariance to find conditions on physical observables
Let's suppose that coordinates have to transform accoring to the Inhomogenous Galilean Group. Then
$$ x' = x + a + v(t+b) $$
$$ t' = t + b $$
Let's use a funtion $\psi(x,t)$ of $x$ and $t$ as the ...
- 372
8
votes
4
answers
2k
views
Doesn't Newton's equation of motion have a bigger invariance group than the Galilean group?
Newton's equation ${F}^i=m\frac{d^2x^i}{dt^2}$ is unchanged in form, under the Galilean group:
(i) under a translation of the origin of coordinates,
(ii) rotation of coordinates, and
(iii) Galilean ...
- 12.3k
1
vote
1
answer
247
views
What is the relationship between the Galilean group and the Poincaré group?
What is the relationship between the Galilean group and the Poincaré group?
Are they siblings within the Lie group? Or does the Poincaré group contain the Galilean group as a subgroup?
I'm not so much ...
- 61
8
votes
0
answers
164
views
What is the symmetry group of Mach's spacetime?
Newtonian spacetime can be modeled as a geometric object $M$ (affine space or manifold with connection with an absolute time function etc. etc.) that is symmetric under the action of the Galilean ...
- 381
2
votes
0
answers
137
views
What do these Casimir invariants of the Galilean group physically represent?
There exist Casimir invariants of the Galilean group which commute with all the generators of the group. They are, of course, Galilean scalars (i.e., scalars under space and time translations, ...
- 12.3k
1
vote
1
answer
292
views
Can we write the mass $M$, a Casimir invariant of the Galilean group, as a function of its generators?
According to Wikipedia, the mass $M$ is one of the Casimir invariants of the Galilean group. Casimir invariants of a group are made out of the generators, and they commute with all the generators of ...
- 12.3k
1
vote
0
answers
38
views
Is Galilean Conformal Algebra (GCA) isomorphic to any other algebra in $d$ dimension?
I was recently studying stuffs related to Conformal Field theory and its Galilean version. It's known that CFT algebra in $d$ dimension is isomorphic to $SO(d+1,1)$ algebra. We also know that Galilean ...
2
votes
1
answer
134
views
Time evolution of Galilean boost
I was introduced the generator of Galilean boost $K=mx-pt$.
I was given an Hamiltonian with several particles: $H=\sum_i \frac{p_i^2}{2m_i}+V(|x_i-x_j|)$ where the potential only depends on the ...
- 197
5
votes
2
answers
445
views
Inönü-Wigner contraction of Poincaré $\oplus$ $\mathfrak{u}$(1)
Metric = (-+++), complex $i$'s are ignored.
Using the following decompositions of the Poincaré generators,
I can write the Poincaré algebra as
I can get the Galilei algebra using the following ...
- 1,081
0
votes
1
answer
76
views
Is there a Group that covers (classical) relative velocities?
I'm not very well versed in Abstract algebra and group theory, so this question might not make sense to begin with, but I got an idea when reading up on how to rigorously calculate relative velocities....
- 163
5
votes
2
answers
274
views
Do total derivatives have anything to do with central extensions?
I recently got interested in the Galilean group and its central extension and found a paper "Quantization on a Lie group: Higher-order Polarizations" by Aldaya, Guerrero and Marmo.
Before asking my ...
- 11.8k
3
votes
1
answer
301
views
Differences between the conformal group and the Schrödinger group?
Facts:
The Maxwell (free) equations (4d) are invariant under the 15 dimensional conformal group.
The free Schrödinger equation in 3d is invariant under the 15 dimensional group "called" Schrödinger ...
- 6,727
2
votes
2
answers
255
views
Unlike rotation, why a $3\times 3$ translation matrix cannot be written in 3D? or can it be?
The effect of rotation in 3d on a vector, $\vec{r}=x\hat{x}=y\hat{y}+z\hat{z}$ is given in the form a matrix product:$$\vec{r}\to O\vec{r}$$ where $O$ is a $3\times3$ proper orthogonal matrix. Can we ...
- 12.3k
5
votes
3
answers
356
views
Silly question about Galilei Group
I have an silly doubt about Galilei Group.
From Wikipedia:
"The Galilean symmetries can be uniquely written as the composition of a rotation, a translation and a uniform motion of spacetime. Let x ...
- 3,149
0
votes
1
answer
590
views
Why is the Galilean group not commutative?
As I understand it, the Galilean transformation is a matrix
$$
\left[ {\begin{array}{ccccc}
R_{11} & R_{12} & R_{13} & v_x & a_x\\
R_{21} & R_{22} & R_{23} & v_y ...
- 4,231
1
vote
2
answers
891
views
Question on Galilean transformation
Let $a$ be a scalar, $D$ a rotation matrix and $b$ and $v$ are $1\times 3$-vectors.
We had the following Galiean transformation:
$(t, x(t)) \to (t + a, Dx + b + v\cdot t)$
But why is it not
$(t, x(...
user102683
1
vote
1
answer
422
views
Galilean group transformations
My problem is the following:
I have difficulties in answering questions (c), (d) and (e).
For (c) my answer was $\sqrt{x^{2}+t^{2}}$ and yes, the group forms the group of all isometries since the ...
1
vote
1
answer
2k
views
Do Galilean boosts and Lorentz boosts share the same generators?
Gottfried and Yan's Quantum Mechanics introduces a generator $N$, called the boost, which generates Galileo transformations. I think in other terminology one might say $N$ generates Galilean boosts, ...
- 416
9
votes
1
answer
4k
views
Every Galilean transformation can be written as the composition of rotation, translation, and uniform motion
Having heard many good things about Arnold's Mathematical Methods of Classical Mechanics, I picked it up and started going through it. While I think I understand all of the definitions he makes, the ...
- 663
4
votes
1
answer
3k
views
How can the Gallilean transformations form a group?
In class my professor said the Galilean transformations form a group of order 10.
$$
x'=x-vt\\
y'=y\\
z'=z\\
t'=t\\
$$
But how do these form a group? I don't see 10 things to interpret as elements. I ...
- 241
6
votes
2
answers
140
views
Possible mechanics based on the known symmetries in the nature (investigating rumor)
Somewhere I've heard about a relative new mathematical result regarding mechanics. Specifically, there is a list of the known symmetries of mechanics (both Newtonian and relativistic), i.e. different ...
- 8,338
0
votes
1
answer
102
views
Proper notation when working with three Euclidean spatial coordinates in a setting with a time parameter
The How does the Euclidean metric is the symmetry group of Euclidean space. It includes rotations and translations.
Say I consider an Euclidean space and a time parameter. How does the Euclidean ...
- 8,693