In
theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ...
, a super-Poincaré algebra is an extension of the
Poincaré algebra to incorporate
supersymmetry
Supersymmetry is a Theory, theoretical framework in physics that suggests the existence of a symmetry between Particle physics, particles with integer Spin (physics), spin (''bosons'') and particles with half-integer spin (''fermions''). It propo ...
, a relation between
boson
In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0, 1, 2, ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have half odd-intege ...
s and
fermion
In particle physics, a fermion is a subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin (spin 1/2, spin , Spin (physics)#Higher spins, spin , etc.) and obey the Pauli exclusion principle. These particles i ...
s. They are examples of
supersymmetry algebras (without
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 ...
s or internal symmetries), and are
Lie superalgebras. Thus a super-Poincaré algebra is a
Z2-graded vector space with a graded Lie bracket such that the even part is a
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 ident ...
containing the Poincaré algebra, and the odd part is built from
spinor
In geometry and physics, spinors (pronounced "spinner" IPA ) are elements of a complex numbers, complex vector space that can be associated with Euclidean space. A spinor transforms linearly when the Euclidean space is subjected to a slight (infi ...
s on which there is an
anticommutation relation with values in the even part.
Informal sketch
The Poincaré algebra describes the isometries of
Minkowski spacetime
In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of gravitation. It combines inertial space and time manifolds into a four-dimensional model.
The model helps show how a s ...
. From the
representation theory of the Lorentz group
The Lorentz group is a Lie group of symmetries of the spacetime of special relativity. This group can be realized as a collection of matrix (mathematics), matrices, linear transformations, or unitary operators on some Hilbert space; it has a v ...
, it is known that the Lorentz group admits two inequivalent complex spinor representations, dubbed
and
.
[The barred representations are conjugate linear while the unbarred ones are complex linear. The numeral refers to the dimension of the representation space. Another more common notation is to write and respectively for these representations. The general irreducible representation is then , where are half-integral and correspond physically to the spin content of the representation, which ranges from in integer steps, each spin occurring exactly once.] Taking their
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of ...
, one obtains
; such decompositions of tensor products of representations into
direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently but analogously for different kinds of structures. As an example, the direct sum of two abelian groups A and B is anothe ...
s is given by the
Littlewood–Richardson rule.
Normally, one treats such a decomposition as relating to specific particles: so, for example, the
pion
In particle physics, a pion (, ) or pi meson, denoted with the Greek alphabet, Greek letter pi (letter), pi (), is any of three subatomic particles: , , and . Each pion consists of a quark and an antiquark and is therefore a meson. Pions are the ...
, which is a
chiral
Chirality () is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek language, Greek (''kheir''), "hand", a familiar chiral object.
An object or a system is ''chiral'' if it is dist ...
vector particle, is 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 nucleus, atomic nuclei ...
-anti-quark pair. However, one could also identify
with Minkowski spacetime itself. This leads to a natural question: if Minkowski space-time belongs to the
adjoint representation, then can Poincaré symmetry be extended to the
fundamental representation? Well, it can: this is exactly the super-Poincaré algebra. There is a corresponding experimental question: if we live in the adjoint representation, then where is the fundamental representation hiding? This is the program of
supersymmetry
Supersymmetry is a Theory, theoretical framework in physics that suggests the existence of a symmetry between Particle physics, particles with integer Spin (physics), spin (''bosons'') and particles with half-integer spin (''fermions''). It propo ...
, which has not been found experimentally.
History
The super-Poincaré algebra was first proposed in the context of the
Haag–Łopuszański–Sohnius theorem, as a means of avoiding the conclusions of the
Coleman–Mandula theorem. That is, the Coleman–Mandula theorem is a no-go theorem that states that the Poincaré algebra cannot be extended with additional symmetries that might describe the
internal symmetries of the observed physical particle spectrum. However, the Coleman–Mandula theorem assumed that the algebra extension would be by means of a commutator; this assumption, and thus the theorem, can be avoided by considering the anti-commutator, that is, by employing anti-commuting
Grassmann number
In mathematical physics, a Grassmann number, named after Hermann Grassmann (also called an anticommuting number or supernumber), is an element of the exterior algebra of a complex vector space. The special case of a 1-dimensional algebra is known a ...
s. The proposal was to consider a
supersymmetry algebra, defined as the
semidirect product
In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. It is usually denoted with the symbol . There are two closely related concepts of semidirect product:
* an ''inner'' sem ...
of a
central extension of the super-Poincaré algebra by a compact
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 ident ...
of internal symmetries.
Definition
The simplest supersymmetric extension of the Poincaré algebra contains two
Weyl spinors with the following anti-commutation relation:
:
and all other anti-commutation relations between the ''Q''s and ''P''s vanish. The operators
are known as supercharges. In the above expression
are the generators of translation and
are the
Pauli matrices
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices that are traceless, Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () ...
. The index
runs over the values
A dot is used over the index
to remind that this index transforms according to the inequivalent conjugate spinor representation; one must never accidentally contract these two types of indexes. The Pauli matrices can be considered to be a direct manifestation of the
Littlewood–Richardson rule mentioned before: they indicate how the tensor product
of the two spinors can be re-expressed as a vector. The index
of course ranges over the space-time dimensions
It is convenient to work with
Dirac spinor
In quantum field theory, the Dirac spinor is the spinor that describes all known fundamental particles that are fermions, with the possible exception of neutrinos. It appears in the plane-wave solution to the Dirac equation, and is a certain comb ...
s instead of Weyl spinors; a Dirac spinor can be thought of as an element of
; it has four components. The
Dirac matrices
In mathematical physics, the gamma matrices, \ \left\\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra \ \mathr ...
are thus also four-dimensional, and can be expressed as direct sums of the Pauli matrices. The tensor product then gives an algebraic relation to the
Minkowski metric
In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of general_relativity, gravitation. It combines inertial space and time manifolds into a four-dimensional model.
The model ...
which is expressed as:
:
and
:
This then gives the full algebra
:
which are to be combined with the normal
Poincaré algebra. It is a closed algebra, since all
Jacobi identities are satisfied and can have since explicit matrix representations. Following this line of reasoning will lead to
supergravity
In theoretical physics, supergravity (supergravity theory; SUGRA for short) is a modern field theory that combines the principles of supersymmetry and general relativity; this is in contrast to non-gravitational supersymmetric theories such as ...
.
Extended supersymmetry
It is possible to add more supercharges. That is, we fix a number which by convention is labelled
, and define supercharges
with
These can be thought of as many copies of the original supercharges, and hence satisfy
:
:
and
:
but can also satisfy
:
and
:
where
is the ''central charge''.
Super-Poincaré group and superspace
Just as the Poincaré algebra generates the
Poincaré group of isometries of Minkowski space, the super-Poincaré algebra, an example of a Lie super-algebra, generates what is known as a
supergroup. This can be used to define
superspace
Superspace is the coordinate space of a theory exhibiting supersymmetry. In such a formulation, along with ordinary space dimensions ''x'', ''y'', ''z'', ..., there are also "anticommuting" dimensions whose coordinates are labeled in Grassmann num ...
with
supercharges: these are the right
coset
In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
s of the Lorentz group within the
super-Poincaré group.
Just as
has the interpretation as being the generator of spacetime translations, the charges
, with
, have the interpretation as generators of superspace translations in the 'spin coordinates' of superspace. That is, we can view superspace as the direct sum of Minkowski space with 'spin dimensions' labelled by coordinates
. The supercharge
generates translations in the direction labelled by the coordinate
By counting, there are
spin dimensions.
Notation for superspace
The superspace consisting of Minkowski space with
supercharges is therefore labelled
or sometimes simply
.
SUSY in 3 + 1 Minkowski spacetime
In Minkowski spacetime, the
Haag–Łopuszański–Sohnius theorem states that the SUSY algebra with N spinor generators is as follows.
The even part of the
star Lie superalgebra is the direct sum of the
Poincaré algebra and a
reductive Lie algebra ''B'' (such that its self-adjoint part is the tangent space of a real compact
Lie group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.
A manifold is a space that locally resembles Eucli ...
). The odd part of the algebra would be
:
where
and
are specific representations of the Poincaré algebra. (Compared to the notation used earlier in the article, these correspond
and
, respectively, also see the footnote where the previous notation was introduced). Both components are conjugate to each other under the * conjugation. ''V'' is an ''N''-dimensional complex representation of ''B'' and ''V''
* is its
dual representation
In mathematics, if is a group and is a linear representation of it on the vector space , then the dual representation is defined over the dual vector space as follows:
: is the transpose of , that is, = for all .
The dual representation ...
. The Lie bracket for the odd part is given by a symmetric
equivariant
In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry group, ...
pairing on the odd part with values in the even part. In particular, its reduced intertwiner from