From Classical Mechanics to Quantum Field Theory

18 From Classical Mechanics to Quantum Field Theory. A Tutorial

1.2.2.1 Vector and tensor fields

A differentiable mapt→ψ(t), withψ(0) =ψ, gives a curve inHpassing through
the pointψ, so that we can define the tangent vector to the curve atψ∈Hto be
given by: (dψ(t)/dt)|t=0. Then a vector field Γ is given once we assign a tangent
vector at every pointψ∈Hin a smooth way, i.e. we give a smooth global section
ofTH:

Γ:H→TH;ψ→(ψ,φ),ψ∈H,φ∈TψH≈H. (1.57)

Also, Γ (ψ) denotes the vector field evaluated at the pointψ∈H, with tangent
vectorφ∈Hatψas given by Eq. (1.57).
Vector fields are derivations on the algebra of functions and we can define the
Lie derivative along Γ of a functionfas follows:

(LΓ(f)) (ψ)=d

dt

f(ψ(t))|t=0. (1.58)

We can introduce local coordinates by choosing an orthonormal basis{ei},sothat
vectors (and tangent vectors) will be represented byn-tuples of complex numbers
ψ=

(

ψ^1 ,···,ψn

)

withψj≡〈ej|ψ〉.Thuswehave^12 :

(LΓ(f)) (ψ)=φi(ψ)

∂f

∂ψi(ψ). (1.59)

Summation over repeated indexes is understood here and in the following.
In the following, we consider examples of:

(i) constant vector fields, given byφ=const.in the second argument of Eq.

(1.57), and originating the one-parameter group:

Rt→ψ(t)=ψ+tφ; (1.60)

(ii) linear vector fields, withφ(ψ) being a linear and homogeneous function

ofψ:φ=Aψ, for some linear operatorA.Inthiscase:

ψ(t)=exp{tA}ψ. (1.61)

Example 1.2.9. Dilation vector field and linear structure.
The dilation vector field is defined as:

Δ:ψ→(ψ,ψ), (1.62)

which corresponds toA=Iin ii) above. In this case Eq. (1.61) becomes:

ψ(t)=etψ. (1.63)

(^12) Asψjis complex:ψj=qj+ipj,qj,pj∈R,the derivative here has to be understood simply
as:∂/∂ψj=∂/∂qj−i∂/∂pj(see also later on).