Types of quantum field thories
We proceed in three steps: first we recall relevant facts about the gauge field itself, then we discuss charged particles in gauge fields, and finally we add spin.
(see: the main article about Yang-Mills theory)
A -principal bundle over .
Above we have used the following notation:
where denotes the covariant derivative.
A gauge transformation is a smooth bundle morphism .
Let be a gauge transformation.
If is a connection on , then is another connection on .
One can identify with a smooth map , namely by , i.e. for all .
The pullback along a gauge transformation restricts to an automorphism of . In terms of the associated map , we have
The Yang-Mills action functional is gauge-invariant, i.e.
for all gauge transformations .
We have and . Under the isomorphism this corresponds to . Since the bilinear form is -invariant by assumption,
for an electromagnetic field theory given by a -bundle over , we have , so that and . In particular, .
Since is abelian, and so on .
Thus, the Yang-Mills equations reduce to Maxwell’s equations for an electromagnetic field on :
Let be a gauge group. A matter type for is a tuple consisting of:
The action functional is gauge invariant, i.e.
for all gauge transformations .
One calculates that , where is the smooth map associated to via . Further, . A computation shows
where . Now we compute
Since is invariant, the invariance of the first term follows. The invariance of the second term is clear.
One can either keep a connection fixed and consider
as a matter field in an “external gauge field”, or consider the combined action functional
(Scalar particle in an external, trivial gauge field)
We consider , , so that necessarily . A scalar field is field for of matter type where and . The action functional is
The Euler-Lagrange equation is the Klein-Gordon equation
where is the Laplace operator and is the exterior coderivative.
(Charged particle in an electromagnetic field, e.g. a -meson)
Let be a -principal bundle over . A field of charge is a field for of matter type , where is defined by and . The action functional is
The Euler-Lagrange equation is covariant Klein-Gordon equation
where is the covariant Laplace operator and is the exterior covariant coderivative.
The Klein-Gordon equations found above are – unlike the Schrödinger equation – of second order on time. Dirac’s motivation was to find a first order equation which upon iteration yields the Klein-Gordon equation. We first discuss free spinors (where free means that they are not coupled to an electromagnetic field, but still feel the “gravity” of the spacetime manifold), and then add the coupling.
We denote by the Clifford algebra on , i.e. the quotient of the tensor algebra of by the ideal generated by , where is the Minkowski scalar product of signature .
The map extends to an anti-automorphism , whose eigenspace decomposition yields the usual -grading on .
We have .
We denote by the group of linear maps that preserve the product . We define
Then, we define a group homomorphism by . This gives a central extension
We denote by the complexification of the Clifford algebra. The bilinear form on extends to a sesquilinear form on defined by .
Multiplication in restricts to an action of on . One can decompose into copies of a subrepresentation :
The representation is isometric with respect to . If is odd, , and is irreducible. If is even, , and with irreducible and .
We also need some facts about spin structures.
A spin structure on is a principal -bundle over together with a bundle morphism such that for all and all .
Let be the Levi-Cevita connection on . Then,
is a connection on .
Finally, we recall the definition of the Dirac operator.
Let be a representation, with . Note that in the above realization of the group , the representation restricts to a representation of . The spinor bundle is the vector bundle .
Clifford multiplication is a map
It is defined as follows. We write with and . We consider the orthonormal frame . Then, and
One can show using above-listed properties of the Clifford algebra that this definition does not depend on the choice of the representative .
The Dirac operator is
where runs over a local orthonormal basis.
Let be a spacetime with spin structure , and considered as a as a Yang-Mills theory over . A free spinor is a field for of type , where , the scalar product is
and is the restriction of the multiplication in to . The action functional is
The Euler-Lagrange equation determined by the action functional is the Dirac equation
We assume spacetime to have even dimension. Weyl spinors have , with the sign corresponding to left/right-handed spinors. Thus, . Further (they are massless). In the standard model, neutrinos are left-handed Weyl spinors.
We assume spacetime to have signature . Dirac spinors have , so that . The function is taken to be . In the standard model, electrons are Dirac spinors.
In the physical literature, the picture is slightly different: The representation space of a spinor is not a subspace of the Clifford algebra, but rather . One can think about this as a further association of to the Clifford bundle using a representation of on . Below we describe this in the case of the electron, i.e. and . Another difference is here that instead of , physicists often use the (non-canonically) isomorphic group .
One starts with the following representation . Consider the -linear map
It satisfies . The Clifford algebra has a universal property that implies that extends uniquely to a representation of . The images of the standard basis vectors are often called -matrices, .
The restriction of the representation to is a representation . It splits into a direct sum of two representations equivalent to and . Using , one checks using the above definition of the group homomorphism that
The above mentioned identification between and is
Under this isomorphism, becomes
Under this identification, the splitting of into a direct sum yields the defining representation, often called and its conjugate, often called .
Finally, the bilinear form becomes
If one can take the trivial spin structure . It has a canonical global section, so that a spinor can be identified with a map . The Dirac operator is now , where . Now, the Dirac equation is
Let’s go back to Dirac’s orginial motivation. Dirac was looking for a first order differential equation
for functions , whose solutions are automatically solutions of the Klein-Gordon equation
where . If is a solution to the first equation,
This is the Klein-Gordon equation, if . This can only be satisfied for matrices, so that better . Since is a symmetric matrix, this can be written as
The smallest matrices satisfying this relation are the above “gamma matrices”. If , there is a solution in dimension two, the “Pauli matrices”.
For a general spacetime , and unlike in the previous remark, . Rather, is given by the Lichnerowiz formula which has in fact been proved first by Schrödinger. So, Dirac’s motivation actually fails for “curved spacetimes”.
Consider principal -bundles over , for . The fibre product is a principal -bundle over , denoted .
If and are connections on and , respectively, then
is a connection on .
Suppose is a vector space and are representations, such that for all and . Then, is a representation of on .
Let be a spacetime with spin structure and let be a Yang-Mills theory over with gauge group . For a representation with , and a commuting representation, the associated bundle still has a Clifford multiplication. For a connection on , one can define a Dirac operator
Let be a spacetime with spin structure, let be a Yang-Mills theory with gauge group over , and let be a representation of on commuting with . A charged spinor is a field for of type , where and is given as before. Its action functional is
(Spinor in an electromagnetic field)
Here, is some representation, and is given by complex multiplication with , where is the charge of the spinor. Obviously and commute. The Euler-Lagrange equation is
If one can take . The canonical global section identifies with a smooth function and the connection with a 1-form with components . Then,
This gives the “Dirac equation?” one usually finds in a textbook.
|gravity||electroweak and strong nuclear force||fermionic matter||scalar field|
|field content:||vielbein field||principal connection||spinor||scalar field|
|Lagrangian:||scalar curvature density||field strength squared||Dirac operator component density||field strength squared + potential density|
Useful literature on this topic is:
Christian Bär, Introduction to Spin Geometry, Oberwolfach Reports 53 (2006), p. 3135-3136.
D. Bleecker, Gauge Theory and Variational Principles, Addison-Weasley, 1981.
H. Blaine Lawson Jr. , Marie-Louise Michelson, Spin geometry, Princeton Univ. Press, 1989.
G. L. Naber, Topology, Geometry and Gauge Dields, Springer, 1999.