Representation theory of the Galilean group
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nonrelativistic The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena in ...
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
, an account can be given of the existence of
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
and spin (normally explained in
Wigner's classification In mathematics and theoretical physics, Wigner's classification is a classification of the nonnegative ~ (~E \ge 0~)~ energy irreducible unitary representations of the Poincaré group which have either finite or zero mass eigenvalues. (Since thi ...
of relativistic mechanics) in terms of the representation theory of the
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 rotati ...
, which is the spacetime symmetry group of nonrelativistic quantum mechanics. In dimensions, this is the subgroup of the
affine group In mathematics, the affine group or general affine group of any affine space over a field is the group of all invertible affine transformations from the space into itself. It is a Lie group if is the real or complex field or quaternions. Rela ...
on (), whose linear part leaves invariant both the metric () and the (independent) dual metric (). A similar definition applies for dimensions. We are interested in
projective representation In the field of representation theory in mathematics, a projective representation of a group ''G'' on a vector space ''V'' over a field ''F'' is a group homomorphism from ''G'' to the projective linear group \mathrm(V) = \mathrm(V) / F^*, where G ...
s of this group, which are equivalent to
unitary representation In mathematics, a unitary representation of a group ''G'' is a linear representation π of ''G'' on a complex Hilbert space ''V'' such that π(''g'') is a unitary operator for every ''g'' ∈ ''G''. The general theory is well-developed in case ''G ...
s of the nontrivial central extension of the universal covering group of the
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 rotati ...
by the one-dimensional Lie group , cf. the article
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 rotati ...
for the central extension of its Lie algebra. The method of
induced representation In group theory, the induced representation is a representation of a group, , which is constructed using a known representation of a subgroup . Given a representation of '','' the induced representation is, in a sense, the "most general" represe ...
s will be used to survey these. We focus on the (centrally extended, Bargmann) Lie algebra here, because it is simpler to analyze and we can always extend the results to the full Lie group through the Frobenius theorem. : ,P_i0 : _i,P_j0 : _,E0 : _i,C_j0 : _,L_i\hbar delta_L_-\delta_L_-\delta_L_+\delta_L_/math> : _,P_ki\hbar delta_P_j-\delta_P_i/math> : _,C_ki\hbar delta_C_j-\delta_C_i/math> : _i,Ei\hbar P_i : _i,P_ji\hbar M\delta_ ~. 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 ...
), ''Pi'' is the generator of translations ( momentum operator), ''Ci'' is the generator of Galilean boosts, and ''Lij'' 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
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 elemen ...
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 operato ...
. The mass-shell invariant :ME- 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 operato ...
. In dimensions, a third
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 operato ...
is , where :\vec \equiv M \vec + \vec\times\vec ~, somewhat analogous to 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 describ ...
of relativistic mechanics. More generally, in dimensions, invariants will be a function of :W_ = M L_ + P_i C_j - P_j C_i and :W_ = P_i L_ + P_j L_ + P_k L_~, as well as of the above mass-shell invariant and central charge. Using
Schur's lemma In mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if ''M'' and ''N'' are two finite-dimensional irreducible representations of a group ' ...
, in an irreducible unitary representation, all these Casimir invariants are multiples of the identity. Call these coefficients and and (in the case of dimensions) , respectively. Recalling that we are considering unitary representations here, we see that these eigenvalues have to be
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s. Thus, , and . (The last case is similar to the first.) In dimensions, when In , we can write, for the third invariant, where represents the spin, or intrinsic angular momentum. More generally, in dimensions, the generators and will be related, respectively, to the total angular momentum and center-of-mass moment by :W_ = M S_ :L_ = S_ + X_i P_j - X_j P_i :C_i = M X_i - P_i t ~. From a purely representation-theoretic point of view, one would have to study all of the representations; but, here, we are only interested in applications to quantum mechanics. There, represents the
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...
, which has to be bounded below, if thermodynamic stability is required. Consider first the case where is nonzero. Considering the (, ) space with the constraint mE = mE_0 + ~, we see that the Galilean boosts act transitively on this hypersurface. In fact, treating the energy as the Hamiltonian, differentiating with respect to , and applying Hamilton's equations, we obtain the mass-velocity relation . The hypersurface is parametrized by this velocity In . Consider the stabilizer of a point on the
orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as ...
, (), where the velocity is . Because of transitivity, we know the unitary
irrep In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W,W ...
contains a nontrivial linear subspace with these energy-momentum eigenvalues. (This subspace only exists in a
rigged Hilbert space In mathematics, a rigged Hilbert space (Gelfand triple, nested Hilbert space, equipped Hilbert space) is a construction designed to link the distribution and square-integrable aspects of functional analysis. Such spaces were introduced to study s ...
, because the momentum spectrum is continuous.) The subspace is spanned by , , and . We already know how the subspace of the irrep transforms under all operators but the
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
. Note that the rotation subgroup is
Spin(3) In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when ) :1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1. As a Li ...
. We have to look at its double cover, because we are considering projective representations. This is called the
little group In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism g ...
, a name given by
Eugene Wigner Eugene Paul "E. P." Wigner ( hu, Wigner Jenő Pál, ; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his co ...
. His method of induced representations specifies that the irrep is given by the direct sum of all the
fiber Fiber or fibre (from la, fibra, links=no) is a natural or artificial substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials. The strongest engineering materials often incorpora ...
s in a
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
over the hypersurface, whose fibers are a unitary irrep of . is none other than . (See
representation theory of SU(2) In the study of the representation theory of Lie groups, the study of representations of SU(2) is fundamental to the study of representations of semisimple Lie groups. It is the first case of a Lie group that is both a compact group and a non-abel ...
, where it is shown that the unitary irreps of are labeled by , a non-negative integer multiple of one half. This is called spin, for historical reasons.) *Consequently, for , the unitary irreps are classified by , and a spin . *Looking at the spectrum of , it is evident that if is negative, the spectrum of is not bounded below. Hence, only the case with a positive mass is physical. *Now, consider the case . By unitarity, mE - = is nonpositive. Suppose it is zero. Here, it is also the boosts as well as the rotations that constitute the little group. Any unitary irrep of this little group also gives rise to a projective irrep of the Galilean group. As far as we can tell, only the case which transforms trivially under the little group has any physical interpretation, and it corresponds to the no-particle state, the
vacuum A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or " void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often di ...
. The case where the invariant is negative requires additional comment. This corresponds to the representation class for = 0 and non-zero . Extending the
bradyon The physics technical term massive particle refers to a massful particle which has real non-zero rest mass (such as baryonic matter), the counter-part to the term massless particle. According to special relativity, the velocity of a massive particl ...
, luxon,
tachyon A tachyon () or tachyonic particle is a hypothetical particle that always travels faster than light. Physicists believe that faster-than-light particles cannot exist because they are not consistent with the known laws of physics. If such partic ...
classification from the representation theory of the Poincaré group to an analogous classification, here, one may term these states as ''synchrons''. They represent an instantaneous transfer of non-zero momentum across a (possibly large) distance. Associated with them, by above, is a "time" operator :t=- ~, which may be identified with the time of transfer. These states are naturally interpreted as the carriers of instantaneous action-at-a-distance forces. N.B. In the -dimensional Galilei group, the boost generator may be decomposed into :\vec = - \vect~, with playing a role analogous to helicity.


See also

*
Galilei-covariant tensor formulation 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 ...
* Representation theory of the Poincaré group *
Wigner's classification In mathematics and theoretical physics, Wigner's classification is a classification of the nonnegative ~ (~E \ge 0~)~ energy irreducible unitary representations of the Poincaré group which have either finite or zero mass eigenvalues. (Since thi ...
*
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 describ ...
* Representation theory of the diffeomorphism group * Rotation operator


References

* Bargmann, V. (1954). "On Unitary Ray Representations of Continuous Groups", ''Annals of Mathematics'', Second Series, 59, No. 1 (Jan., 1954), pp. 1–46 * . * * Gilmore, Robert (2006). ''Lie Groups, Lie Algebras, and Some of Their Applications'' (Dover Books on Mathematics) {{DEFAULTSORT:Representation Theory Of The Galilean Group Rotational symmetry Quantum mechanics Representation theory of Lie groups