Mathematical Principles of Theoretical Physics

3.1. BASIC CONCEPTS 115

1) For a real (complex) scalar fieldφ, the associated vector bundle isM⊗pR^1 (M⊗pC):

φ:M→M⊗pR^1 (M⊗pC).

2) LetAμbe a 4-dimensional vector field. Then

Aμ:M→TM,

and for anyp∈M,TpMis the Minkowski space.

3) For a 4-dimensional covector fieldAμ, we have

Aμ:M→T∗M,

and for anyp∈M,Tp∗Mis the dual space ofTpM.

4) A(k,r)-tensor fieldTonM:

T={Tνμ 11 ······νμrk},

is expressed by

T:M→TrkM,

whereTrkMis the(k,r)-tensor bundle onM, denoted by

(3.1.25) TrkM=T︸M⊗ ··· ⊗︷︷ TM︸

k

⊗︸T∗M⊗ ··· ⊗︷︷ T∗M︸

r

,

and⊗represents the tensor product.

5) The Dirac spinor fieldΨis defined by

Ψ:M→M⊗pC^4.

In particular, forNDirac spinor fieldsΨ= (ψ^1 ,···,ψN)T,

Ψ:M→M⊗p(C^4 )N.

6) The Riemann metricgμ νdefined on a 4D space-time manifoldM, representing the

gravitational potential, is a mapping:

gμ ν:M→T 20 M,

whereT 20 M=T∗M⊗T∗Mis a (0,2)-tensor bundle as defined by (3.1.25).

The fields given by 1)-6) above include all types of physical fields, and the associated
vector bundles are physically significant.
In classical theories of interactions, the physical fields and the associated bundle spaces
are given as follows:

1) Gravity:

(3.1.26) gμ ν:M→T∗M⊗T∗M.