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A.3 Momentum Space

We consider a cartesian momentum space in coordinates (px,py,pz) along axes (^x,^y,^z). The vector position is momentum space is written

P = px ^x+ py^y + pz^z
(A.208)

We consider the two following curvilinear systems:

A.3.1 System ( ) p∥,p⊥,φ

Definition

The coordinates ( ) p∥,p⊥,φ are defined on the space

-∞≤ p∥ < ∞ (A.209)
0 ≤ p⊥ < ∞ (A.210)
0 ≤ φ < 2π (A.211)

and is related to (px,py,pz) by

p∥ = pz (A.212)
p⊥ = ∘ ------- p2x + p2y
φ = arctan(py∕px) + πH(- px) [2π]

which is inverted to

px = p⊥cosφ (A.213)
py = p⊥sinφ
pz = p∥

Position Vector

The position vector in momentum space then becomes

P = p⊥⊥^+ p ∥^∥
(A.214)

where we define a local orthonormal basis ( ) ^∥, ^⊥, ^φ as

^∥ = ^z (A.215)
^ ⊥ = cosφ^x + sinφ^y (A.216)
^φ = ^p ∥×^p ⊥ = -sinφ^x + cosφy^ (A.217)

Covariant Basis

The covariant vector basis is defined in (A.1), which becomes here

e∥ = -∂P- ∂p ∥ = ^∥ (A.218)
e⊥ = ∂P ∂p-- ⊥ = ^⊥ (A.219)
eφ = ∂P- ∂φ = p⊥∂^⊥- ∂φ = p⊥^φ (A.220)

so that we have the covariant basis

( ) ( ) ^ ^ e ∥,e⊥, eφ = ∥,⊥,p⊥ ^φ
(A.221)

the scaling factors

(h ,h ,h ) = (1,1,p ) ∥ ⊥ φ ⊥
(A.222)

and the normalized tangent basis

( ) (^e ,^e ,^e ) = ^∥, ^⊥,φ^ ∥ ⊥ φ
(A.223)

Contravariant Basis

The Contravariant vector basis is defined in (A.9), which becomes here

e∥ = ∇p ∥ = ^∥ (A.224)
e⊥ = ∇p ⊥ = ^⊥ (A.225)
eφ = ∇φ = φ^ --- p⊥ (A.226)

The relations (A.10-A.11) are here readily verified. The normalized reciprocal basis is

( ) ( ) ^e∥,^e⊥,^eφ = ^∥, ^⊥, ^φ
(A.227)

which here coincides with the normalized tangent basis, since both bases are orthogonal.

Metric Coefficients

They are defined in (A.12) and become here

gij = ( ) 1 0 0 ( 0 1 0 ) 0 0 p2 ⊥ (A.228)
gij = ( ) 1 0 0 ( 0 1 0 ) 0 0 1∕p2⊥

As a result

2 g = p⊥
(A.229)

and the Jacobian is

J = p⊥
(A.230)

Differential elements

dl( ) p∥ = dp∥ (A.231)
dl(p⊥) = dp⊥ (A.232)
dl(φ) = p⊥dφ (A.233)

dS( ) p∥ = p⊥dp⊥dφ^∥ (A.234)
dS(p⊥) = p⊥dp∥dφ^ ⊥ (A.235)
dS(φ) = dp∥dp⊥^φ (A.236)
3 d X = p⊥dp ∥dp⊥dφ
(A.237)

Christoffel Symbols

They are defined in (A.49) and are all zero here except

{ } { } φ φ 1-φφ∂g-φφ -1- p⊥ R = R p⊥ = 2g ∂p⊥ = p⊥

{ } p⊥ 1- RR ∂gφφ- φ φ = -2 g ∂p⊥ = - p⊥
(A.238)

Differential Operations

Gradient

∂f ∂f 1 ∂f ∇pf = ---^∥ + ----^⊥ + ------^φ ∂p∥ ∂p⊥ p⊥ ∂φ
(A.239)

Divergence

( ) ( ) ∇ ⋅A = -∂-- A ⋅^∥ + 1---∂-- p A ⋅ ^⊥ + -1--∂-(A ⋅ ^φ) p ∂p ∥ p⊥ ∂p⊥ ⊥ p⊥ ∂ φ
(A.240)

Curl

(∇p × A ) ⋅^∥ = -1- p⊥-∂-- ∂p⊥(p⊥A ⋅φ^) --1- p⊥-∂- ∂ φ( ) A ⋅ ^⊥ (A.241)
(∇p × A ) ⋅^ ⊥ = -1- p⊥∂-- ∂φ( ^ ) A ⋅∥--∂-- ∂p∥ (A ⋅φ^) (A.242)
(∇p × A ) ⋅^φ = -∂-- ∂p∥( ) A ⋅ ^⊥--∂-- ∂p⊥( ) A ⋅^∥ (A.243)

A.3.2 System (p,ξ,φ )

Definition

The coordinates (p,ξ,φ) are defined on the space

0 ≤ p < ∞ (A.244)
- 1 ≤ ξ < 1 (A.245)
0 ≤ φ < 2π (A.246)

and is related to (px,py,pz) by

p = ∘ ------------ p2x + p2y + p2z (A.247)
ξ = p ∘-----z------- p2x + p2y + p2z (A.248)
φ = arctan(py∕px) + πH(- px) [2π]

which is inverted to

px = p∘ ------ 1 - ξ2 cosφ (A.249)
py = p∘1----ξ2 sinφ
pz = pξ

Note that we have the following transformation from (p ,p ) ∥ ⊥ to (p,ξ)

p = ∘ ------- p2∥ + p2⊥ (A.250)
ξ = p ∘----∥--- p2∥ + p2⊥

which is inverted to

p∥ = pξ (A.251)
p⊥ = p∘ ------ 1 - ξ2

Position Vector

The position vector in momentum space then becomes

P = p^p
(A.252)

where we define a local orthonormal basis ( ^ ) ^p,ξ, ^φ as

^p = ∘ ------ 1 - ξ2(cosφ^x + sin φ^y) + ξ^z (A.253)
^ξ = ^φ×^p = ξ(cosφ^x + sinφy^) -∘ ------ 1- ξ2z^ (A.254)
^φ = -sinφ^x + cosφy^ (A.255)

Covariant Basis

The covariant vector basis is defined in (A.1), which becomes here

ep = ∂P- ∂p = ^p (A.256)
eξ = ∂P- ∂ξ = p∂^p- ∂ξ = ----p---- ∘ ----2- 1- ξ^ ξ (A.257)
eφ = ∂P- ∂φ = p∂^p- ∂φ = p ------ ∘ 1 - ξ2^φ (A.258)

so that we have the covariant basis

( ) ( ) p ∘ ------ ep,eξ,e φ = ^p,- ∘------2^ξ,p 1- ξ2φ^ 1- ξ
(A.259)

the scaling factors

( ) (h ,h ,h ) = 1,- ∘---p---,p∘1----ξ2 p ξ φ 1 - ξ2
(A.260)

and the normalized tangent basis

( ) (^e ,^e ,^e ) = ^p, ^ξ, ^φ p ξ φ
(A.261)

Contravariant Basis

The Contravariant vector basis is defined in (A.9), which becomes here

ep = ∇p = ∂p- ∂P = ^p (A.262)
eξ = ∇ξ = ∂ξ- ∂P = -∘ ------ --1--ξ2- p ^ ξ (A.263)
eφ = ∇φ = ∂ φ --- ∂P = 1 -∘------- p 1 - ξ2^φ (A.264)

The relations (A.10-A.11) are here readily verified. The reciprocal basis is

( ∘ ------ ) ( p ξ φ) --1---ξ2^ ----1---- e ,e ,e = ^p,- p ξ,p∘1----ξ2 ^φ
(A.265)

and the normalized reciprocal basis is

( p ξ φ) ( ) ^e ,^e ,^e = ^p, ^ξ,φ^
(A.266)

which here coincides with the normalized tangent basis, since both bases are orthogonal.

Metric Coefficients

They are defined in (A.12) and become here

gij = ( ) 1 0 ( ) 0 ( 0 p2∕ 1- ξ2 0 ( ) ) 0 0 p2 1 - ξ2 (A.267)
gij = ( ) 1 0( 2) 2 0 ( 0 1- ξ ∕p 0 [ ( )] ) 0 0 1∕ p2 1- ξ2

As a result

g = p4
(A.268)

and the Jacobian is

J = p2
(A.269)

Differential elements

dl(p) = dp (A.270)
dl(ξ) = ∘--p---- 1- ξ2dξ (A.271)
dl(φ ) = p∘ ------ 1- ξ2dφ (A.272)

dS(p) = p2dξdφ^p (A.273)
dS(ξ) = -p∘ -----2 1 - ξdpdφ^ ξ (A.274)
dS(φ ) = ∘--p---- 1 - ξ2dpdξ^φ (A.275)
d3X = p2dpd ξdφ
(A.276)

Christoffel Symbols

They are defined in (A.49) and are all zero here except

Differential Operations

Gradient

∘ ------ ∂f 1 - ξ2∂f 1 ∂f ∇pf = ∂p-^p- ----p----∂ξ ^ξ +-∘------2∂φ-^φ p 1 - ξ
(A.277)

Divergence

( ) ∇ ⋅A = 1--∂-(p2A ⋅p^) - 1-∂-- ∘1---ξ2A ⋅ ^ξ + -∘-1-----∂--(A ⋅ ^φ) p p2∂p p ∂ξ p 1- ξ2 ∂φ
(A.278)

Curl

(∇ × A) ⋅^p = 1- p∂-- ∂ξ( ∘ ------ ) 1- ξ2A ⋅ ^φ + -∘--1---- p 1 - ξ2-∂- ∂φ( ) A ⋅ ^ξ (A.279)
(∇ × A) ⋅^ξ = 1 p-∂ ∂p- (pA ⋅ ^φ) - 1 -∘-----2- p 1- ξ∂ ∂φ-(A ⋅ ^p)
(∇ × A) ⋅^φ = -1- p-∂- ∂p( ^) pA ⋅ξ-∘ ------ --1--ξ2- p∂(A-⋅p^)- ∂ ξ

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