The Galilei-covariant tensor formulation is a method for treating non-relativistic physics using the extended Galilei group as the representation group of the theory. It is constructed in the light cone of a five dimensional manifold.
Takahashi et al., in 1988, began a study of
Galilean symmetry
In physics, a Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics. These transformations together with spatial rotatio ...
, where an explicitly covariant non-relativistic field theory could be developed. The theory is constructed in the light cone of a (4,1)
Minkowski space
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inerti ...
.
Previously, in 1985, Duval et al. constructed a similar tensor formulation in the context of
Newton–Cartan theory Newton–Cartan theory (or geometrized Newtonian gravitation) is a geometrical re-formulation, as well as a generalization, of Newtonian gravity first introduced by Élie Cartan and Kurt Friedrichs and later developed by Dautcourt, Dixon, Dombrowski ...
. Some other authors also have developed a similar Galilean tensor formalism.
Galilean manifold
The Galilei transformations are
:
where
stands for the three-dimensional Euclidean rotations,
is the relative velocity determining Galilean boosts, a stands for spatial translations and b, for time translations. Consider a free mass particle
; the mass shell relation is given by
.
We can then define a 5-vector,
:
,
with
.
Thus, we can define a scalar product of the type
:
where
:
is the metric of the space-time, and
.
Extended Galilei algebra
A five dimensional
Poincaré algebra leaves the metric
invariant,
:
We can write the generators as
:
The non-vanishing commutation relations will then be rewritten as
:
An important Lie subalgebra is
:
is the generator of time translations (
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
), ''P
i'' is the generator of spatial translations (
momentum operator
In quantum mechanics, the momentum operator is the operator (physics), operator associated with the momentum (physics), linear momentum. The momentum operator is, in the position representation, an example of a differential operator. For the case o ...
),
is the generator of Galilean boosts, and
stands for a generator of rotations (
angular momentum operator
In quantum mechanics, the angular momentum operator is one of several related operators analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic and molecular physics and other quantum prob ...
). The generator
is a
Casimir invariant
In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum operator ...
and
is an additional
Casimir invariant
In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum operator ...
. This algebra is isomorphic to the extended
Galilean Algebra in (3+1) dimensions with
, The
central charge
In theoretical physics, a central charge is an operator ''Z'' that commutes with all the other symmetry operators. The adjective "central" refers to the center of the symmetry group—the subgroup of elements that commute with all other element ...
, interpreted as mass, and
.
The third Casimir invariant is given by
, where
is a 5-dimensional analog of the
Pauli–Lubanski pseudovector
In physics, the Pauli–Lubanski pseudovector is an operator defined from the momentum and angular momentum, used in the quantum-relativistic description of angular momentum. It is named after Wolfgang Pauli and Józef Lubański,
It describe ...
.
Bargmann structures
In 1985 Duval, Burdet and Kunzle showed that four-dimensional Newton–Cartan theory of gravitation can be reformulated as
Kaluza–Klein reduction of five-dimensional Einstein gravity along a null-like direction. The metric used is the same as the Galilean metric but with all positive entries
:
This lifting is considered to be useful for non-relativistic
holographic
Holography is a technique that enables a wavefront to be recorded and later re-constructed. Holography is best known as a method of generating real three-dimensional images, but it also has a wide range of other Holography#Applications, applic ...
models. Gravitational models in this framework have been shown to precisely calculate the mercury precession.
See also
*
Galilean group
In physics, a Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics. These transformations together with spatial rotatio ...
*
Representation theory of the Galilean group
In nonrelativistic quantum mechanics, an account can be given of the existence of mass and spin (normally explained in Wigner's classification of relativistic mechanics) in terms of the representation theory of the Galilean group, which is the ...
*
Lorentz group
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicis ...
*
Poincaré group
The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our und ...
*
Pauli–Lubanski pseudovector
In physics, the Pauli–Lubanski pseudovector is an operator defined from the momentum and angular momentum, used in the quantum-relativistic description of angular momentum. It is named after Wolfgang Pauli and Józef Lubański,
It describe ...
References
{{DEFAULTSORT:Galilean Covariance
Theoretical physics
Rotational symmetry
Quantum mechanics
Representation theory of Lie groups