The topic today in this group theory thread is “sixtors and representations of the Lorentz group”.
Consider the group of proper orthochronous Lorentz transformations and the transformation law of the electromagnetic tensor . The components of this antisymmetric tensor can be transformed into a sixtor or and we can easily write how the Lorentz group acts on this 6D vector ignoring the spacetime dependence of the field.
Under spatial rotations, transform separately in a well-known way giving you a reducible representation of the rotation subgroup in the Lorent orthochronous group. Remember that rotations are a subgroup of the Lorentz group, and it contains Lorentz boosts in additionto those rotations. In fact, in the space of sixtors and they are thus a reducible representation, a direct sum group representation. That is, rotations leave invariant subspaces formed by and invariant. However, these two subspaces mix up under Lorentz boosts! We have written before how transform under general boosts but we can simplify it without loss of generality and for some matrices . So it really seems that the representation is “irreducible” under the whole group. But it is NOT true! Irreducibility does not hold if we ALLOW for COMPLEX numbers as coefficients for the sixtors/bivectors (so, it is “tricky” and incredible but true: you change the numbers and the reducibility or irreducibility character does change. That is a beautiful connection betweeen number theory and geometry/group theory). It is easy to observe that using the Riemann-Silberstein vector
and allowing complex coefficients under Lorent transformations, such that
i.e., it transforms totally SEPARATELY from each other () under rotations and the restricted Lorentz group. However, what we do have is that using complex coefficients (complexification) in the representation space, the sixtor decomposes into 2 complex conjugate 3 dimensional representaions. These are irreducible already, so for rotations alone transformations are complex orthogonal since if you write
with and . Be aware: here is an imaginary angle. Moreover, transforms as follows from the following equation:
Remark: Rotations in 4D are given by a unitary 4-vector such as and the rotation matrix is given by the general formula
If you look at this rotation matrix, and you assign with , the above rotations are in fact the same transformations of the electric and magnetic parts of the sixtor! Thus the representation of the general orthochronous Lorentz group is secretly complex-orthogonal for electromagnetic fields (with complex coefficients)! We do know already that
are the electromagnetic main invariants. So, complex geometry is a powerful tool too in group theory! :). The real and the imaginary part of this invariant are also invariant. The matrices of 2 subrespresentations formed here belong to the complex orthogonal group . This group is a 3 dimensional from the complex viewpoint but it is 6 dimensional from the real viewpoint. The orthochronous Lorentz group is mapped homomorphically to this group, and since this map has to be real and analytic over the group such that, as Lie groups, . We can also use the complex rotation group in 3D to see that the 2 subrepresentations must be inequivalent. Namely, pick one of them as the definition of the group representation. Then, it is complex analytic and its complex parameter provide any equivalent representation. Moreover, any other subrepresentation is complex conjugated and thus antiholomorphic (in the complex sense) in the complex parameters.
Generally, having a complex representation, i.e., a representation in a COMPLEX space or representation given by complex valued matrices, implies that we get a complex conjugated reprentation which can be equivalent to the original one OR NOT. BUT, if share with original representation the property of being reducible, irreducible or decomposable. Abstract linear algebra says that to any representation in complex vector spaces there is always a complex conjugate representation in the complex conjugate vector space . Mathematically, one ca consider representations in vector spaces over various NUMBER FIELDS. When the number field is extended or changed, irreducibility MAY change into recubibility and vice versa. We have seen that the real sixtor representation of the restricted Lorentz group is irreducible BUT it becomes reducible IF it is complexified! However, its defining representation by real 4-vectors remains irreducible under complexification. In Physics, reducibility is usually referred to the field of complex numbers , since it is generally more beautiful (it is algebraically closed for instance) and complex numbers ARE the ground field of representation spaces. Why is this so? There are two main reasons:
1st. Mathematical simplicity. is an algebraically closed filed and its representation theory is simpler than the one over the real numbers. Real representations are obtained by going backwards and “inverting” the complexification procedure. This process is sometimes called “getting the real forms” of the group from the complex representations.
2nd. Quantum Mechanics seems to prefer complex numbers (and Hilbert spaces) over real numbers or any other number field.
The importance of is understood from the Maxwell equations as well. In vacuum, without sources or charges, the full Maxwell equations read
These equations are Lorentz covariant and reducibility is essential there. It is important to note that
implies that we can choose ONLY one of the components of the sixtor, or , or one single component of the sixtor is all that we need. If in the induction law there were a plus sign instead of a minus sign, then both representations could be used simultaneously! Furthermore, Lorentz covariance would be lost! Then, the Maxwell equations in vacuum should satisfy a Schrödinger like equation due to complex linear superposition principle. That is, if and are solutions then a complex solution with complex coefficients should also be a solution. This fact would imply invariance under the so-called duality transformation
However, it is not true due to the Nature of Maxwell equations and the (apparent) absence of isolated magnetic charges and currents!
The sixtor or 6D Riemann-Silberstein vector is a complex-valued quantity up to one multiplicative constant and it can be understood as a bivector field in Clifford algebras/geometric calculus/geometric algebra. But we are not going to go so far in this post. We remark only that a bivector field is something different to a normal vector or even, as we saw in the previous post, a bivector field can not be the same thing that a spinor field with spin. Moreover, the electric and magnetic parts of the sixtor transform as vectors under spatial rotations
where and R being an orientation preserving rotation matrix in the spcial orthogonal group . Remember that
The group is bases on the preservation of the scalar (inner) product defined by certain quadratic form acting on 3D vectors:
and so for proper rotations. This excludes spatial reflections or parity transformations P, that in fact are important too. Parity transformations act differently to electric and magnetic fields and they have . Parity transformations belong to the group of “improper” rotations in 3D space.
However, the electromagnetic field components are NOT related to the spatial components of a 4D vector. That is not true. With respect to the proper Lorentz group:
and where the metric is the Minkovski metric. In fact, the explicit representation of
by its matrix elements
Despite this fact, that the electromagnetic field sixtor transforms as a vector under the COMPLEX special orthogonal group in 3D space
This observation is related to the fact that the proper Lorentz group and the complex rotation group are isomorphic to each other as Lie groups, i.e. . This analogy and mathematical result has some deeper consequences in the theory of the so-called Dirac, Weyl and Majorana spinors (quantum fields describing fermions with differnt number of independent “components”) in the massive case.
The puzzle, now, is to understand why the mass term is forbidden in
as it is the case of the electromagnetic (classical) field. Moreover, in this problem, we will see that there is a relation between symmetries and operators of the Lie groups and the corresponding generators of their respective Lie algebras. Let’s begin with pure boosts in some space-time plane:
and where we defined, as usual,
and , with .
and where we have defined the boost generator as in the plane , and the boost parameter will be given by the number
Remark: Lorentz transformations/boosts corresponds to rotations with an “imaginary angle”.
Moreover, we also get
Equivalently, by “empathic mimicry” we can introduce the boost generators in the remaining 3 planes as follows:
In addition to these 3 Lorentz boosts, we can introduce and define another 3 generators related to the classicla rotation in 3D space. Their generators would be given by:
Therefore, and span the proper Lorent Lie algebra with generators . These generators satisfy the commutators:
with and the totally antisymmetric tensor. This Levi-Civita symbol is also basic in the structure constants. Generally speaking, in the Physics realm, the generators are usually chosen to be hermitian and an additional imaginar factor should be included in the above calculations to get hermitian generators. If we focus on the group over the real numbers, i.e., the usual rotation group, the Lie algebra basis is given by , or equivalently by the matrices
and the commutation rules are
If the rotation matrix is approximated to:
then we have that the rotation matrix is antisymmetric, since we should have
so the matrix generators are antisymmetric matrices in the Lie algebra. That is, the generators are real and antisymmetric. The S-matrices span the full real Lie algebra. In the same way, we could do the same for the complex group and we could obtain the Lie algebra over the complex numbers with added to the real of the real Lie group. This defines a “complexification” of the Lie group and it implies that:
Do you remember this algebra but with another different notation? Yes! This is the same Lie algebra we obtained in the case of the Lorentz group. Therefore, the Lie algebras of and are isomorphic if we make the identifications:
The rotation part of is the rotation part of
The boost part of is the complex conjugated (rotation-like) part of
Then for every matrix we have
and for every matrix we have
For instance, a bivector boost in some axis provides:
and where in the first matrix, it acts on , the second matrix (as the first one) acts also on a complex sixtor, and where the rotation around the axis perpendicular to the rotation plane is defined by the matrix operator:
and this matrix would belong to the real orthogonal group .
Note: Acting onto the sixtor as a bivector field shows that it generates the correct Lorentz transformation of the full electromagnetic field! The check is quite straightforward, since
From this complex matrix we easily read off the transformation of electric and magnetic fields:
Note the symmetry between electric and magnetic fields hidden in the sixtor/bivector approach!
For the general Lorentz transformation we have the invariants
In the next group theory threads we are going to study the relationship between Special Relativity, electromagnetic fields and the complex group .
There is a close interdependence of the following three concepts:
The classical electromagnetic fields and can be in fact combined into a complex six dimensional (6D) vector, sometimes called SIXTOR or Riemann-Silberstein vector:
and where the numerical prefactor is conventional ( you can give up for almost every practical purposes).
Moreover, we have
where and so
and where we have used natural units for simplicity.
The Maxwell-Faraday equation reads:
The Ampère circuital law in vacuum reads:
These two equations can be combined into a single equation using the Riemann-Silberstein vector or sixtor :
B) Comparing both sides in A), we easily get and
We can take the divergence of the time derivative of the sixtor:
Therefore, and hold in the absence of electric and magnetic charges on any section of a Minkovski spacetime, and everywhere! The presence of electric charges and the absence of magnetic charges, the so-called magnetic monopoles, breaks down the gauge symmetry of
Introducing 3 matrices with the aid of the 3D Levi-Civita tensor , the completely antisymmetric tensor with 3 indices such that and we can write these matrices as follows:
If for , then
We can define matrices so
and then for . Experts in Clifford/geometric algebras will note that these matrices are in fact “Dirac matrices” up to a conventional sign.
In fact, you can admire the remarkable similarity between the sixtor equation AND the Dirac equation as follows:
In summary: the sixtor equation is a Dirac-like equation (but of course the electromagnetic field is not a fermion!).
The equation for , since , will be the feynmanity
Let us define the formal adjoint field and the 4 components of a “density-like” quantity
Then, we can recover the classical result that says that the energy density and the Poynting vector of the electromagnetic field is
These equations provide an important difference between the Dirac equation for a massive spin true (anti)particle and the electromagnetic massless spin photon, because you can observe that in the former case you HAVE:
and you HAVE
in the latter (the electromagnetic field has not mass term!). In fact, you also have that for a Dirac field the current is defined to be:
and it transforms like a VECTOR field under Lorentz transformations, while the previous current are the components of some stress-energy-momentum !!!! They are NOT the same thing!
In fact, transform under the and (complex conjugated) representation of the proper Lorentz group.
Remark: Belinfante coined the term “undor” when dealing with fields transforming according to some specific representations of the Lorentz group.