Superspace
Encyclopedia
"Superspace" has had two meanings in physics. The word was first used by John Wheeler
to describe the configuration space
of general relativity
; for example, this usage may be seen in his famous 1973 textbook Gravitation
.
The second meaning refers to 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 number
s rather than real numbers. The ordinary space dimensions correspond to boson
ic degrees of freedom, the anticommuting dimensions to fermionic degrees of freedom.
See also supermanifold
(although the definition of a superspace as a supermanifold here does not agree with the definition used in that article).
Rm|n is the Z2-graded vector space
with Rm as the even subspace and Rn as the odd subspace. The same definition applies to Cm|n.
extending in n real, bosonic directions and no fermionic directions. The vector space R0|n, which is the n-dimensional real Grassmann algebra. The space R1|1 of one even and one odd direction is known as the space of dual number
s, introduced by William Clifford
in 1873.
with N supercharge
s is often formulated in the superspace R1|2N, which contains one real direction t identified with time
and N complex Grassmann directions which are spanned by Θi and Θ*i, where i runs for 1 to N.
Consider the special case N = 1. The superspace R1|2 is a 3-dimensional vector space. A given coordinate therefore may be written as a triple (t, Θ, Θ*). The coordinates form a Lie superalgebra
, in which the gradation degree of t is even and that of Θ and Θ* is odd. This means that a bracket may be defined between any two elements of this vector space, and that this bracket reduces to the commutator
on two even coordinates and on one even and one odd coordinate while it is an anticommutator on two odd coordinates. This superspace is an abelian Lie superalgebra, which means that all of the forementioned brackets vanish
where
is the commutator of a and b and 
is the anticommutator of a and b.
One may define functions from this vectorspace to itself, which are called superfield
s. The above algebraic relations imply that, if we expand our superfield as a power series in Θ and Θ* then we will only find terms at the zeroeth and first orders, because Θ2 = Θ*2 = 0. Therefore superfields may be written as arbitrary functions of t multiplied by the zeroeth and first order terms in the two Grassmann coordinates
Superfields, which are representations of the supersymmetry
of superspace, generalize the notion of tensor
s, which are representations of the rotation group of a bosonic space.
One may then define derivatives in the Grassmann directions, which take the first order term in the expansion of a superfield to the zeroeth order term and annihilate the zeroeth order term. One can choose sign conventions such that the derivatives satisfy the anticommutation relations
These derivatives may be assembled into supercharge
s
whose anticommutators identify them as the fermionic generators of a supersymmetry
algebra
where i times the time derivative is the Hamiltonian
operator in quantum mechanics
. Both Q and its adjoint anticommute with themselves. The supersymmetry variation with supersymmetry parameter ε of a superfield Φ is defined to be
We can evaluate this variation using the action of Q on the superfields
Similarly one may define covariant derivative
s on superspace
which anticommute with the supercharges and satisfy a wrong sign supersymmetry algebra
The fact that the covariant derivatives anticommute with the supercharges means the supersymmetry transformation of a covariant derivative of a superfield is equal to the covariant derivative of the same supersymmetry transformation of the same superfield. Thus, generalizing the covariant derivative in bosonic geometry which constructs tensors from tensors, the superspace covariant derivative constructs superfields from superfields.
is R4|4, which is the direct sum
of four real bosonic dimensions and four real Grassmann dimensions. In supersymmetric quantum field theories
one is interested in superspaces which furnish representations
of a Lie superalgebra
called a supersymmetry algebra
. The bosonic part of the supersymmetry algebra is the Poincaré algebra, while the fermionic part is constructed using spinor
s of Grassmann numbers.
For this reason, in physical applications one considers an action of the supersymmetry algebra on the four fermionic directions of R4|4 such that they transform as a spinor
under the Poincaré subalgebra. In four dimensions there are three distinct irreducible 4-component spinors. There is the Majorana spinor, the left-handed Weyl spinor and the right-handed Weyl spinor. The CPT theorem
implies that in a unitary
, Poincaré invariant theory, which is a theory in which the S-matrix is a unitary matrix and the same Poincaré generators act on the asymptotic in-states as on the asymptotic out-states, the supersymmetry algebra must contain an equal number of left-handed and right-handed Weyl spinors. However, since each Weyl spinor has four components, this means that if one includes any Weyl spinors one must have 8 fermionic directions. Such a theory is said to have extended supersymmetry, and such models have received a lot of attention. For example, supersymmetric gauge theories with eight supercharges and fundamental matter have been solved by Nathan Seiberg
and Edward Witten
, see Seiberg–Witten gauge theory. However in this subsection we are considering the superspace with four fermionic components and so no Weyl spinors are consistent with the CPT theorem.
Note: There are many sign convention
s in use and this is only one of them.
This leaves us with one possibility, the four fermionic directions transform as a Majorana spinor θα. We can also form a conjugate spinor
where C is the charge conjugation matrix, which is defined by the property that when it conjugates a gamma matrix, the gamma matrix is negated and transposed. The first equality is the definition of while the second is a consequence of the Majorana spinor condition θ* = iγ0Cθ. The conjugate spinor plays a role similar to that of θ* in the superspace R1|2, except that the Majorana condition, as manifested in the above equation, imposes that θ and θ* are not independent.
In particular we may construct the supercharges
which satisfy the supersymmetry algebra
where
is the 4-momentum
operator. Again the covariant derivative is defined like the supercharge but with the second term negated and it anticommutes with the supercharges. Thus the covariant derivative of a supermultiplet is another supermultiplet.
is sometimes used variously as a term to refer to the multiverse
, a form of hyperspace
/subspace
, or the substrate of the omniverse
.
John Archibald Wheeler
John Archibald Wheeler was an American theoretical physicist who was largely responsible for reviving interest in general relativity in the United States after World War II. Wheeler also worked with Niels Bohr in explaining the basic principles behind nuclear fission...
to describe the configuration space
Configuration space
- Configuration space in physics :In classical mechanics, the configuration space is the space of possible positions that a physical system may attain, possibly subject to external constraints...
of general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
; for example, this usage may be seen in his famous 1973 textbook Gravitation
Gravitation (book)
In physics, Gravitation is a very important reference book on Einstein's theory of gravity by Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler. Often considered the "Bible" of General Relativity by researchers for its prominence. It is frequently called MTW after its authors' initials....
.
The second meaning refers to the coordinate space of a theory exhibiting supersymmetry
Supersymmetry
In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...
. In such a formulation, along with ordinary space dimensions x, y, z, ..., there are also "anticommuting" dimensions whose coordinates are labeled in Grassmann number
Grassmann number
In mathematical physics, a Grassmann number, named after Hermann Grassmann, is a mathematical construction which allows a path integral representation for Fermionic fields...
s rather than real numbers. The ordinary space dimensions correspond to boson
Boson
In particle physics, bosons are subatomic particles that obey Bose–Einstein statistics. Several bosons can occupy the same quantum state. The word boson derives from the name of Satyendra Nath Bose....
ic degrees of freedom, the anticommuting dimensions to fermionic degrees of freedom.
See also supermanifold
Supermanifold
In physics and mathematics, supermanifolds are generalizations of the manifold concept based on ideas coming from supersymmetry. Several definitions are in use, some of which are described below.- Physics :...
(although the definition of a superspace as a supermanifold here does not agree with the definition used in that article).
Rm|n is the Z2-graded vector space
Graded vector space
In mathematics, a graded vector space is a type of vector space that includes the extra structure of gradation, which is a decomposition of the vector space into a direct sum of vector subspaces.-N-graded vector spaces:...
with Rm as the even subspace and Rn as the odd subspace. The same definition applies to Cm|n.
Trivial examples
The smallest superspace is a point which contains neither bosonic nor fermionic directions. Other trivial examples include the n-dimensional real plane Rn, which is a vector spaceVector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
extending in n real, bosonic directions and no fermionic directions. The vector space R0|n, which is the n-dimensional real Grassmann algebra. The space R1|1 of one even and one odd direction is known as the space of dual number
Dual number
In linear algebra, the dual numbers extend the real numbers by adjoining one new element ε with the property ε2 = 0 . The collection of dual numbers forms a particular two-dimensional commutative unital associative algebra over the real numbers. Every dual number has the form z = a + bε with a and...
s, introduced by William Clifford
William Clifford
William Clifford was an English cricketer of the mid-19th century who played for Kent...
in 1873.
The superspace of supersymmetric quantum mechanics
Supersymmetric quantum mechanicsSupersymmetric quantum mechanics
In theoretical physics, supersymmetric quantum mechanics is an area of research where mathematical concepts from high-energy physics are applied to the seemingly more prosaic field of quantum mechanics.-Introduction:...
with N supercharge
Supercharge
In theoretical physics, a supercharge is a generator of supersymmetry transformations.Supercharge, denoted by the symbol Q, is an operator which transforms bosons into fermions, and vice versa...
s is often formulated in the superspace R1|2N, which contains one real direction t identified with time
Time
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....
and N complex Grassmann directions which are spanned by Θi and Θ*i, where i runs for 1 to N.
Consider the special case N = 1. The superspace R1|2 is a 3-dimensional vector space. A given coordinate therefore may be written as a triple (t, Θ, Θ*). The coordinates form a Lie superalgebra
Lie superalgebra
In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2-grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry...
, in which the gradation degree of t is even and that of Θ and Θ* is odd. This means that a bracket may be defined between any two elements of this vector space, and that this bracket reduces to the commutator
Commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:...
on two even coordinates and on one even and one odd coordinate while it is an anticommutator on two odd coordinates. This superspace is an abelian Lie superalgebra, which means that all of the forementioned brackets vanish
where
is the commutator of a and b and is the anticommutator of a and b.One may define functions from this vectorspace to itself, which are called superfield
Superfield
In theoretical physics, one often analyzes theories with supersymmetry in which superfields play a very important role. In four dimensions, the simplest example - namely the minimal N=1 supersymmetry - may be written using a superspace with four extra fermionic coordinates...
s. The above algebraic relations imply that, if we expand our superfield as a power series in Θ and Θ* then we will only find terms at the zeroeth and first orders, because Θ2 = Θ*2 = 0. Therefore superfields may be written as arbitrary functions of t multiplied by the zeroeth and first order terms in the two Grassmann coordinates
Superfields, which are representations of the supersymmetry
Supersymmetry
In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...
of superspace, generalize the notion of tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
s, which are representations of the rotation group of a bosonic space.
One may then define derivatives in the Grassmann directions, which take the first order term in the expansion of a superfield to the zeroeth order term and annihilate the zeroeth order term. One can choose sign conventions such that the derivatives satisfy the anticommutation relations
These derivatives may be assembled into supercharge
Supercharge
In theoretical physics, a supercharge is a generator of supersymmetry transformations.Supercharge, denoted by the symbol Q, is an operator which transforms bosons into fermions, and vice versa...
s
-
-
and 
-
whose anticommutators identify them as the fermionic generators of a supersymmetry
Supersymmetry
In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...
algebra
where i times the time derivative is the Hamiltonian
Hamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
operator in quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
. Both Q and its adjoint anticommute with themselves. The supersymmetry variation with supersymmetry parameter ε of a superfield Φ is defined to be
We can evaluate this variation using the action of Q on the superfields
Similarly one may define covariant derivative
Covariant derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given...
s on superspace
-
-
and 
-
which anticommute with the supercharges and satisfy a wrong sign supersymmetry algebra
-
-
.
-
The fact that the covariant derivatives anticommute with the supercharges means the supersymmetry transformation of a covariant derivative of a superfield is equal to the covariant derivative of the same supersymmetry transformation of the same superfield. Thus, generalizing the covariant derivative in bosonic geometry which constructs tensors from tensors, the superspace covariant derivative constructs superfields from superfields.
Four-dimensional N = 1 superspace
Perhaps the most popular superspace in physicsPhysics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
is R4|4, which is the direct sum
Direct sum of modules
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The result of the direct summation of modules is the "smallest general" module which contains the given modules as submodules...
of four real bosonic dimensions and four real Grassmann dimensions. In supersymmetric quantum field theories
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
one is interested in superspaces which furnish representations
Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
of a Lie superalgebra
Lie superalgebra
In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2-grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry...
called a supersymmetry algebra
Supersymmetry algebra
In theoretical physics, a super-Poincaré algebra is an extension of the Poincaré algebra to incorporate supersymmetry, a relation between bosons and fermions. They are examples of supersymmetry algebras, and hence are Lie superalgebras...
. The bosonic part of the supersymmetry algebra is the Poincaré algebra, while the fermionic part is constructed using spinor
Spinor
In mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...
s of Grassmann numbers.
For this reason, in physical applications one considers an action of the supersymmetry algebra on the four fermionic directions of R4|4 such that they transform as a spinor
Spinor
In mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...
under the Poincaré subalgebra. In four dimensions there are three distinct irreducible 4-component spinors. There is the Majorana spinor, the left-handed Weyl spinor and the right-handed Weyl spinor. The CPT theorem
CPT theorem
In quantum field theory the CPT theorem states that any canonical quantum field theory is invariant under the CPT operation, which is a combination of three discrete transformations: charge conjugation C, parity transformation P, and time reversal T...
implies that in a unitary
Unitary
Unitary may refer to:* Unitary construction, in automotive design, another common term for a unibody or monocoque construction**Unitary as chemical weapons opposite of Binary...
, Poincaré invariant theory, which is a theory in which the S-matrix is a unitary matrix and the same Poincaré generators act on the asymptotic in-states as on the asymptotic out-states, the supersymmetry algebra must contain an equal number of left-handed and right-handed Weyl spinors. However, since each Weyl spinor has four components, this means that if one includes any Weyl spinors one must have 8 fermionic directions. Such a theory is said to have extended supersymmetry, and such models have received a lot of attention. For example, supersymmetric gauge theories with eight supercharges and fundamental matter have been solved by Nathan Seiberg
Nathan Seiberg
Nathan "Nati" Seiberg is an Israeli American theoretical physicist who works on string theory. He was recipient of a 1996 MacArthur Fellowship and the Dannie Heineman Prize for Mathematical Physics in 1998. He is currently a professor at the Institute for Advanced Study in Princeton, New Jersey, USA...
and Edward Witten
Edward Witten
Edward Witten is an American theoretical physicist with a focus on mathematical physics who is currently a professor of Mathematical Physics at the Institute for Advanced Study....
, see Seiberg–Witten gauge theory. However in this subsection we are considering the superspace with four fermionic components and so no Weyl spinors are consistent with the CPT theorem.
Note: There are many sign convention
Sign convention
In physics, a sign convention is a choice of the physical significance of signs for a set of quantities, in a case where the choice of sign is arbitrary. "Arbitrary" here means that the same physical system can be correctly described using different choices for the signs, as long as one set of...
s in use and this is only one of them.
This leaves us with one possibility, the four fermionic directions transform as a Majorana spinor θα. We can also form a conjugate spinor
where C is the charge conjugation matrix, which is defined by the property that when it conjugates a gamma matrix, the gamma matrix is negated and transposed. The first equality is the definition of while the second is a consequence of the Majorana spinor condition θ* = iγ0Cθ. The conjugate spinor plays a role similar to that of θ* in the superspace R1|2, except that the Majorana condition, as manifested in the above equation, imposes that θ and θ* are not independent.
In particular we may construct the supercharges
which satisfy the supersymmetry algebra
where
is the 4-momentumMomentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...
operator. Again the covariant derivative is defined like the supercharge but with the second term negated and it anticommutes with the supercharges. Thus the covariant derivative of a supermultiplet is another supermultiplet.
Superspace in popular culture
Superspace in science fictionScience fiction
Science fiction is a genre of fiction dealing with imaginary but more or less plausible content such as future settings, futuristic science and technology, space travel, aliens, and paranormal abilities...
is sometimes used variously as a term to refer to the multiverse
Multiverse
The multiverse is the hypothetical set of multiple possible universes that together comprise all of reality.Multiverse may also refer to:-In fiction:* Multiverse , the fictional multiverse used by DC Comics...
, a form of hyperspace
Hyperspace (science fiction)
Hyperspace is a plot device sometimes used in science fiction. It is typically described as an alternative region of space co-existing with our own universe which may be entered using an energy field or other device...
/subspace
Hyperspace (science fiction)
Hyperspace is a plot device sometimes used in science fiction. It is typically described as an alternative region of space co-existing with our own universe which may be entered using an energy field or other device...
, or the substrate of the omniverse
Omniverse
The Omniverse is the conceptual ensemble of all possible universes, with all possible laws of physics.In this physical cosmology context, the limitation of the definition of "universe" that it has only one set of "physical laws and constants that govern them," is expanded to include multiple sets...
.