PROBABILITY CURRENT IN
ZERO-SPIN RELATIVISTIC
QUANTUM MECHANICS
Tamás Fülöp
Budapest Univ. Techn. Econ.
Tamás Matolcsi
Eötvös Univ., Budapest
10th December, 2015, Zimányi School, Budapest, Hungary
Klein–Gordon equation for scalar wave function :
no conserving positive definite probability current
only conserving charge current
Feshbach–Villars formalism on
ϕ+
i
m
(∂0 + ieA0 ) ϕ
ϕ−
i
m
(∂0 + ieA0 ) ϕ
efforts to choose the ‘positive’ solutions
Spin 1/2 : ψγ µ ψ , no efforts needed
!
Varadarajan : Geometry
of quantum theory I–II.
(1968–70)
only the free case
only in momentum space
only the representation
coordinate space?
physically interesting investigations?
nonfree extension ?
(first of all : how does his construction work ?)
geometric picture for the spin 1/2 case =⇒ spin 0 case easy
Notations:
~ ≡ c ≡ 1,
g with signature (+−−−) ,
spacetime four-indices: µ, ν = 0, 1, 2, 3
three-indices: j, k = 1, 2, 3
complex conjugate: ∗ ,
m > 0 only,
transposition :
p ∈ Pm : p0 =
p
pj p j +
T
,
m2
combination :
,
R
Pm
d3 p
p0
†
Matrices:
σ1 =
0 1
,
1 0
σ2 =
0
,
−I2
β = γ0 =
I2
0
0
i
−i
,
0
γj =
σ3 =
0
σj
1
0
−σj
,
0
0
,
−1
αj =
I2 =
0
σj
σj
0
1 0
0 1
Properties:
γµ γν + γν γµ = 2gµν ,
αj = −βγj ,
Frequently :
γ0† = γ0 ,
㵆 β = βγµ ,
γj† = −γj ,
αj† = αj
θ ∈ C 4 , Θ ∈ C 4 ⊗ C4 :
θ = θ† β,
Θ = Θ† β
Geometric ingredients:
p ∈ Pm : eigenvalues of pµ γ µ are +m , −m ,
the eigensubspaces: N+ (p) , N− (p) , both 2 dimensional
 
m
0
µ
0
:
For example: p̌ = 
p̌
γ
=
mγ
= mβ ,
µ
0
0
   
 
0
1
0
0 1
0
 ,
 ,  ,
N+ (p̌): 
N
(p̌):
−
0 0
1
0
0
0
αj pj + βm:
+p0 , −p0 ,
N+ (p) , N− (p)
 
0
0
 
0
1
Spin 1/2, free quantum mechanics:
γ µ pµ ψ(p) = mψ(p):
For example:
ψ(p) ∈ N+ (p)
 
m
0
:
p̌ = 
0
0
 
ψ1
ψ2 

ψ=
0
0
To Lorentz transformation L, exists DL :
†
−1
,
βDL
β = DL
4
ψ : Pm → C :
−1
DL (γ µ pµ ) DL
= γ µ (Lp)µ
ipµ aµ
U(a,L) ψ (p) = e
DL ψ(L−1 p)
irreducible ray representation of the Poincaré group
Hilbert space:
R
Pm
md3 p
ψ(p)ψ(p) p0
=
R
†
Pm
ψ(p)
m2 d3 p
ψ(p) p2
0
only the solutions of
γ µ pµ ψ(p) = mψ(p) ,
(αj pj + βm) ψ(p) = p0 ψ(p)
Foldy–Wouthuysen transformation :
ψ W (p) = W (p)ψ(p),
W (p) = √
ψW


ψ1W
 W
ψ2 

=
 0 


0
1
2m(p0 +m)
at any p
[(p0 + m)I4 − αj pj ]
Coordinate space:
Z
Z
µ
e−ipµ x
3
md p
=
ψ(p)
p0
ψ ψ (t, x) d3 x
†
ψ(p) ∈ N+ (p),
Z
3
3
md
p
−ip0 x0 ipj xj
e
= (2π) 2 ψ(t, x)
e
ψ(p)
p0
is time independent, equals the momentum
space norm
γ µ pµ ψ = mψ,
(αj pj + βm) ψ = p0 ψ
=⇒
γ µ (i∂µ )ψ(x) = mψ(x),
i∂t ψ(t, x) = [αj (−i∂j ) + βm] ψ(t, x)
Current and Lagrangian :
ψ † ψ = ψβψ = ψγ 0 ψ :
component of
j µ = ψγ µ ψ
real, positive definite, conserved, Noether current :
[γ µ (i∂µ ) − m] ψ = 0: E–L equation from L = ψ [γ µ (i∂µ ) − m] ψ,
ψ 7→ eiχ ψ :
Remark :
Noether current
✷+m
2
ψ = 0:
Klein–Gordon componentwise
[multiplying by γ µ (i∂µ ) + m ]
Spin 0, free quantum mechanics: analogously
ψ : C4
ζ : C4 ∧ C4
ψ(p) ∈ N+ (p)
ζ(p) ∈ N+ (p) ∧ N+ (p)
γ µ pµ ψ = mψ
γ µ pµ ζ = mζ
R
md3 p
ψψ p0
=
R
ψ
†
m2 d3 p
ψ p2
0
irrep of Poincaré
essentially 2 DOF
R
Tr ζζ
md3 p
p0
=
irrep of Poincaré
R
†
Tr ζ ζ
m2 d3 p
essentially 2 ∧ 2 = 1 DOF
p20
n1 , n2 ∈ N+ (p),
n3 , n4 ∈ N− (p):
(γ µ pµ − m) (c12 n1 ∧ n2 + c13 n1 ∧ n3 + · · · + c34 n3 ∧ n4 ) = 0:
only c12 6= 0
Foldy–Wouthuysen :

ψ1W

 W
ψ 2 

Wψ = 
 0 


0
W ζW
T

0
 W
−ζ12
=
 0

0
W
ζ12
0
0
0
0
0
0
0

0

0


0
0
µ
µ
j = ψγ ψ
µ
µ
µ
j = Tr ζγ ζ
µ
L = ψ [γ (i∂µ ) − m] ψ
L = Tr ζ [γ (i∂µ ) − m] ζ
KG componentwise
KG componentwise
Nonfree quantum mechanics:
∂µ ❀ ∂µ + ieAµ
µ
∂µ Tr ζ 1 γ ζ2 = 0
=⇒
∂t
R
0
Tr ζ 1 γ ζ2 ≡ ∂t
time independent Hilbert space scalar product
µ
µ
2
(∂ + ieA ) (∂µ + ieAµ ) − m ζ 6= 0
( !)
R
Tr
† ζ1 ζ2
= 0:
For the future:
relationship to KG (free, nonfree)
Coulomb problem
second quantization
new types of field theoretical interaction ?
For the future:
relationship to KG (free, nonfree)
Coulomb problem
second quantization
new types of field theoretical interaction ?
THANK YOU FOR YOUR ATTENTION!