The Poincaré group, named after
Henri Poincaré
Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "Th ...
(1906), was first defined by
Hermann Minkowski
Hermann Minkowski (; ; 22 June 1864 – 12 January 1909) was a German mathematician and professor at Königsberg, Zürich and Göttingen. He created and developed the geometry of numbers and used geometrical methods to solve problems in number t ...
(1908) as the
group of
Minkowski spacetime isometries. It is a ten-dimensional
non-abelian Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
that is of importance as a model in our understanding of the most basic fundamentals of
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
.
Overview
A
Minkowski spacetime isometry has the property that the interval between
event
Event may refer to:
Gatherings of people
* Ceremony, an event of ritual significance, performed on a special occasion
* Convention (meeting), a gathering of individuals engaged in some common interest
* Event management, the organization of ev ...
s is left invariant. For example, if everything were postponed by two hours, including the two events and the path you took to go from one to the other, then the time interval between the events recorded by a stop-watch you carried with you would be the same. Or if everything were shifted five kilometres to the west, or turned 60 degrees to the right, you would also see no change in the interval. It turns out that the
proper length
Proper length or rest length is the length of an object in the object's rest frame.
The measurement of lengths is more complicated in the theory of relativity than in classical mechanics. In classical mechanics, lengths are measured based on ...
of an object is also unaffected by such a shift. A time or space reversal (a reflection) is also an isometry of this group.
In Minkowski space (i.e. ignoring the effects of
gravity
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
), there are ten degrees of freedom of the
isometries, which may be thought of as translation through time or space (four degrees, one per dimension); reflection through a plane (three degrees, the freedom in orientation of this plane); or a "
boost" in any of the three spatial directions (three degrees). Composition of transformations is the operation of the Poincaré group, with
proper rotations being produced as the composition of an even number of reflections.
In
classical physics
Classical physics is a group of physics theories that predate modern, more complete, or more widely applicable theories. If a currently accepted theory is considered to be modern, and its introduction represented a major paradigm shift, then the ...
, 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 rota ...
is a comparable ten-parameter group that acts on
absolute time and space
Absolute space and time is a concept in physics and philosophy about the properties of the universe. In physics, absolute space and time may be a preferred frame.
Before Newton
A version of the concept of absolute space (in the sense of a preferr ...
. Instead of boosts, it features
shear mappings to relate co-moving frames of reference.
Poincaré symmetry
Poincaré symmetry is the full symmetry of
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The law ...
. It includes:
* ''
translation
Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
s'' (displacements) in time and space (''P''), forming the
abelian
Abelian may refer to:
Mathematics Group theory
* Abelian group, a group in which the binary operation is commutative
** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms
* Metabelian group, a grou ...
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
of translations on space-time;
* ''
rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
s'' in space, forming the non-Abelian Lie group of
three-dimensional rotations (''J'');
* ''
boosts'', transformations connecting two uniformly moving bodies (''K'').
The last two symmetries, ''J'' and ''K'', together make the
Lorentz group (see also
Lorentz invariance
In a relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation
In physics, the Lorentz transformations are a six-parameter famil ...
); the
semi-direct product
In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product:
* an ''inner'' semidirect product is a particular way in w ...
of the translations group and the Lorentz group then produce the Poincaré group. Objects that are invariant under this group are then said to possess Poincaré invariance or relativistic invariance.
10 generators (in four spacetime dimensions) associated with the Poincaré symmetry, by
Noether's theorem, imply 10 conservation laws: 1 for the energy, 3 for the momentum, 3 for the angular momentum and 3 for the velocity of the center of mass.
Poincaré group
The Poincaré group is the group of Minkowski spacetime
isometries. It is a ten-dimensional
noncompact Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
. The
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
of
translation
Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
s is a
normal subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
, while the
Lorentz group is also a subgroup, the
stabilizer of the origin. The Poincaré group itself is the minimal subgroup of the
affine group which includes all translations and
Lorentz transformation
In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant velo ...
s. More precisely, it is a
semidirect product
In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product:
* an ''inner'' semidirect product is a particular way in wh ...
of the translations and the Lorentz group,
:
with group multiplication
:
.
Another way of putting this is that the Poincaré group is a
group extension of the
Lorentz group by a vector
representation of it; it is sometimes dubbed, informally, as the inhomogeneous Lorentz group. In turn, it can also be obtained as a
group contraction
In theoretical physics, Eugene Wigner and Erdal İnönü have discussed the possibility to obtain from a given Lie group a different (non-isomorphic) Lie group by a group contraction with respect to a continuous subgroup of it. That amounts to a l ...
of the de Sitter group SO(4,1) ~ Sp(2,2), as the
de Sitter radius goes to infinity.
Its positive energy unitary irreducible
representations are indexed by
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 ele ...
(nonnegative number) and
spin
Spin or spinning most often refers to:
* Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning
* Spin, the rotation of an object around a central axis
* Spin (propaganda), an intentionally ...
(
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
or half integer) and are associated with particles 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 chemistry, ...
(see
Wigner's classification).
In accordance with the
Erlangen program, the geometry of Minkowski space is defined by the Poincaré group: Minkowski space is considered as a
homogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements ...
for the group.
In
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, the universal cover of the Poincaré group
:
which may be identified with the double cover
:
is more important, because representations of
are not able to describe fields with spin 1/2; i.e.
fermions. Here
is the group of complex
matrices with unit determinant, isomorphic to the
Lorentz-signature spin group .
Poincaré algebra
The Poincaré algebra is the
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
of the Poincaré group. It is a
Lie algebra extension
In the theory of Lie groups, Lie algebras and their representation theory, a Lie algebra extension is an enlargement of a given Lie algebra by another Lie algebra . Extensions arise in several ways. There is the trivial extension obtained by ...
of the Lie algebra of the Lorentz group. More specifically, the proper (
),
orthochronous (
) part of the Lorentz subgroup (its
identity component),
, is connected to the identity and is thus provided by the
exponentiation
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to ...
of this
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
. In component form, the Poincaré algebra is given by the commutation relations:
where
is the
generator of translations,
is the generator of Lorentz transformations, and
is the
Minkowski metric (see
Sign convention).
The bottom commutation relation is the ("homogeneous") Lorentz group, consisting of rotations,
, and boosts,
. In this notation, the entire Poincaré algebra is expressible in noncovariant (but more practical) language as
:
where the bottom line commutator of two boosts is often referred to as a "Wigner rotation". The simplification
permits reduction of the Lorentz subalgebra to
and efficient treatment of its associated
Representation theory of the Lorentz group, representations. In terms of the physical parameters, we have
:
The
Casimir invariants of this algebra are
and
where
is the
Pauli–Lubanski pseudovector; they serve as labels for the representations of the group.
The Poincaré group is the full symmetry group of any
relativistic field theory
In physics, a field is a physical quantity, represented by a scalar, vector, or tensor, that has a value for each point in space and time. For example, on a weather map, the surface temperature is described by assigning a number to each point ...
. As a result, all
elementary particle
In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include electrons, the fundamental fermions ( quarks, leptons, ...
s fall in
representations of this group. These are usually specified by the ''four-momentum'' squared of each particle (i.e. its mass squared) and the intrinsic
quantum number
In quantum physics and chemistry, quantum numbers describe values of conserved quantities in the dynamics of a quantum system. Quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian—quantities that can ...
s
, where
is the
spin
Spin or spinning most often refers to:
* Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning
* Spin, the rotation of an object around a central axis
* Spin (propaganda), an intentionally ...
quantum number,
is the
parity
Parity may refer to:
* Parity (computing)
** Parity bit in computing, sets the parity of data for the purpose of error detection
** Parity flag in computing, indicates if the number of set bits is odd or even in the binary representation of the ...
and
is the
charge-conjugation quantum number. In practice, charge conjugation and parity are violated by many
quantum field theories
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles ...
; where this occurs,
and
are forfeited. Since
CPT symmetry
Charge, parity, and time reversal symmetry is a fundamental symmetry of physical laws under the simultaneous transformations of charge conjugation (C), parity transformation (P), and time reversal (T). CPT is the only combination of C, P, and T ...
is
invariant in quantum field theory, a
time-reversal quantum number may be constructed from those given.
As a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
, the group has four connected components: the component of the identity; the time reversed component; the spatial inversion component; and the component which is both time-reversed and spatially inverted.
Other dimensions
The definitions above can be generalized to arbitrary dimensions in a straightforward manner. The ''d''-dimensional Poincaré group is analogously defined by the semi-direct product
:
with the analogous multiplication
:
.
The Lie algebra retains its form, with indices and now taking values between and . The alternative representation in terms of and has no analogue in higher dimensions.
Super-Poincaré algebra
A related observation is that the
representations of the Lorentz group include a pair of inequivalent two-dimensional complex
spinor
In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...
representations
and
whose
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
is the
adjoint representation
In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is ...
. One may identify this last bit with four-dimensional Minkowski space itself (as opposed to identifying it with a spin-1 particle, as would normally be done for a pair of
fermion
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
s, e.g. a
pion being composed of a
quark
A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All commonly ...
-anti-quark pair). This strongly suggests that it might be possible to extend the Poincaré algebra to also include spinors. This leads directly to the notion of the
super-Poincaré algebra. The mathematical appeal of this idea is that one is working with the
fundamental representations, instead of the adjoint representations. The physical appeal of this idea is that the fundamental representations correspond to
fermion
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
s, which are seen in nature. So far, however, the implied
supersymmetry
In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories ...
here, of a symmetry between spatial and fermionic directions, has not been seen experimentally in nature. The experimental issue can roughly be stated as the question: if we live in the adjoint representation (Minkowski spacetime), then where is the fundamental representation hiding?
See also
*
Euclidean group
In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space \mathbb^n; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformations ...
*
Representation theory of the Poincaré group
*
Wigner's classification
*
Symmetry in quantum mechanics
*
Center of mass (relativistic) In physics, relativistic center of mass refers to the mathematical and physical concepts that define the center of mass of a system of particles in relativistic mechanics and relativistic quantum mechanics.
Introduction
In non-relativistic phys ...
*
Pauli–Lubanski pseudovector
*
Particle physics and representation theory
There is a natural connection between particle physics and representation theory, as first noted in the 1930s by Eugene Wigner. It links the properties of elementary particles to the structure of Lie groups and Lie algebras. According to thi ...
*
Continuous spin particle
Notes
References
*
*
*
{{DEFAULTSORT:Poincare Group
Lie groups
Group
Quantum field theory
Theory of relativity
Symmetry