Exact solutions of the Dirac equation and induced representations of the Poincare group on the lattice

0.07 MB·English

by  M. Lorente

Exact solutions of the Dirac equation and induced representations of the
Poincar´e group on the lattice
Miguel Lorente
Departamento de F´ısica, Universidad de Oviedo, 33007 Oviedo, Spain
We deduce the structure of the Dirac field on the lattice from the discrete version of differential geom-
etry and from the representation of the integral Lorentz transformations. The analysis of the induced
representations of the Poincar´egroup on the lattice reveals that they are reducible, a result that can be
4 considered a group theoretical approach to the problem of fermion doubling.
0
0
2
1. DISCRETE DIFFERENTIAL and the usual exterior product, from
n
a GEOMETRY which we construct the p-form and the dual
J
(n−p) -form in the n-dimensional space.

9

Given a function of one independent variable
1
v the forward and backward differences are de- σ = 1σ ∆xi1 ∧···∧∆xip
3 fined as p! i1···ip
1
1
0 (∗σ) = σi1···ipε
1 ∆f(x) = f(x+∆x)−f(x) k1···kn−p p! i1···ipk1···kp
0
4 ∇f(x) = f(x)−f(x−∆x)
Fromthep-formswecanconstruct(p+1)-
0
formswith thehelp of theexterior difference,
/
at Similarly we define the average operators for instance,
l
-
p 1
e ∆f(x) = {f(x+∆x)+f(x)}
h 2 ∆ω ≡ ∆a∧∆x+∆b∧∆y =
v: ∇ef(x) = 1 {f(x)+f(x−∆x)} ∆ ∆˜ b ∆˜ ∆ a
x y x y
i 2 − ∆x∧∆y
X ∆x ∆y !
e
r This calculus can be enlarged to functions of
a
The exterior calculus can be used to ex-
several variables. In particular, we have for
press the laws of physics in terms of differ-
the total difference operator
encecalculus, suchasclassical electrodynam-
ics, quantum field theory [1]. In particular,
∆f(x,y)f (x+∆x,y+∆y)−f (x,y) =
for the wave equation we have −∗∆∗∆φ= 0,
∆ ∆ f(x,y)+∆ ∆ f (x,y)
x y x y or
e e
inanobviousnotation. Thefiniteincrements
of the variables ∆x,∆y···, can be used to −∇˜ ∇˜ ∇˜ ∇ ∆˜ ∆˜ ∆˜ ∆ +
x y z t x y z t
introduce a discrete differential form
n +∇ ∇˜ ∇˜ ∇˜ ∆ ∆˜ ∆˜ ∆˜ +···
x y z t x y z t
ω = a(x,y)∆x+b(x,y)∆y . φ(xyzt) = 0 o
2. EXACT SOLUTIONS FOR THE we obtain from the Hamilton equations of
DIRAC-EQUATION motion, the Dirac equation
Given a scalar function Φ(n ) ≡ (n ε )
µ µ µ iγ δµ+ −m cη+ ψ(n ) =0 (3)
µ 0 µ
defined in the grid points of a Minkowski lat-
tice with elementary lengths ε , and the dif- (cid:0) (cid:1)
µ
and from this we recover the Klein-Gordon
ference operators
equation. The transfer matrix, which carries
δ+ ≡ 1 ∆ ∆˜ , δ− ≡ 1 ∇ ∇˜ the Dirac field from one time to the next can
µ εµ µν 6= µ ν µ εµ µν 6= µ ν be obtained from the evolution operator [3]
Y Y
η+ ≡ ∆˜µ, η− ≡ ∇˜µ, µ,ν = 1,2,3,4 U = 1+ 12iε0H , ψ(n +1)= Uψ(n )U†
Yµ Yµ 1− 1iε H 0 0
2 0
the Klein-Gordon equation can be read off
AbasisforthesolutionstotheDiracequa-
δ+δµ− −m2c2η+η− Φ(n ) = 0 (1)
µ 0 µ tion can be obtained from the functions (2)
(cid:16) (cid:17) and the standard four components spinors.
Using the method of separation of vari-
Our model for the fermion fields satisfies the
ables, we obtain the exact solutions of this
following conditions [2]:
difference equation
i) theHamiltonianistranslationalinvariant
3 1− 1iε k nµ
f(nµ) = µ = 0 1+ 221iεµµkµµ! (2) ii) the Hamiltonian is hermitian
Y
iii) form = 0,thewaveequationisinvariant
0
with k , continuous variables satisfiying the
µ under global chiral transformations
dispersion relations k kµ = m2c2
µ 0
Thesolutions(2)constituteabasisforthe iv) there is no fermion doubling
solutionsoftheKlein-Gordonequation. Ifwe
v) the Hamiltonian is non-local
impose spatial boundary conditions on the
wave functions, we get
Finally, coupling the vector field to the
2 πm
i electro-magnetic vector potential we con-
k = tan , m = 0,···,N −1 , i = 1,2,3
i i
εi N struct a gauge invariant vector current lead-
With the help of this basis we can derive, ing to the correct axial anomaly [2]
asinthecontinuumcase,Hamiltonequations
of motion, a Fourier expansion for the Klein-
Gordon fields [2]. We can apply the same
3. INTEGRAL LORENTZ
technique to the Dirac fields on the lattice.
TRANSFORMATIONS
Starting from the Hamiltonian
N−1
H = ε ε ε ∆˜ ∆˜ ∆˜ ψ+(n) × A group theoretical approach to the Dirac
1 2 3 1 2 3
equation, can be given from the induced rep-
n1nX2n3=0
1 1 resentations of Poincar´e group on the lattice
× γ γ ∆ ∆˜ ∆˜ +γ γ ∆˜ ∆ ∆˜ +
0 1ε 1 2 3 0 2 1ε 2 3 [4]. We start with the integral transforma-
(cid:26) 1 2
1 tions of the complete Lorentz group gener-
+ γ γ ∆˜ ∆˜ ∆ +γ ∆˜ ∆˜ ∆˜ ψ
0 3 1 2ε 3 0 1 2 3 (n) ated by the Kac generators
3 (cid:27)
1 0 0 0 1 0 0 0 4. INDUCED REPRESENTATIONS
0 0 1 0 0 1 0 0 OF THE POINCARE´ GROUP
S =  , S =  
1 0 1 0 0 2 0 0 0 1
   
0 0 0 1 0 0 1 0 Given an integral Lorentz transformation
   
1 0 0 0  2 1 1 1 Λ and discrete translation characterized by
0 1 0 0 −1 0 −1 −1 a four-vector a, the induced representations
S =  ,S =  
3 0 0 1 0 4 −1 −1 0 −1 are given by standard methods [5]
   
0 0 0 −1 −1 −1 −1 0 o
    i) chooseanUIR,Dk(a)ofthetranslation
   
Any integral matrix of the complete group (the kernel of the discrete Fourier
Lorentz group can be factorized transform) (5))
L = PηPθPiS ...S PδPεPζS SαSβSγ ii) define the little group H ∈ SO(3,1)
1 2 3 4 4 1 2 3 4 1 2 3 perm. by the stability condition: D(ha) =
n o
D(a),h ∈H
where P = S S S S S , P = S S S ,
1 1 2 3 2 1 2 2 3 2
P3 = S3, α,β,... = 0,1 iii) constructforthegroupT4×sH theUIR
o o
The finite representations of the Lorentz Dk,α(a,h) = Dk(a)⊗Dα(h)
group can be obtained by standard method.
iv) choose coset generators c,c′ such that
In particular, for the two dimensional repre-
(a˜,∧˜)c = c′(a,h). For the coset repre-
sentation α ∈ SL(2,C) and the parity oper-
sentatives we can choose ∧ ≡ L(k) that
ator I we obtain the (reducible) Dirac rep-
s o
resentation [5] takes k into an arbitrary integral vector
of the unit hyperboloid. Finally we have
o
α 0 0 1 Dk,α (a˜,∧˜) =
D(α,Is) =  0 α† −1 ⊗ 1 0! Dkk,′k(′a˜)Dα L−1(k′)∧˜L(k)
 (cid:16) (cid:17) 
(cid:0) (cid:1)
InordertogetUIR,weintroducethepro-
jection operator 5. IRREDUCIBILITY AND
FERMION DOUBLING
1 1
Q(p) = (I +W (p)) , W (p) ≡ γµp
µ
2 m c
0 The dual group of the discrete translation
groupis given by allthe pointsfrom theBril-
Applying this operator to a four compo-
louinzone. Wecharacterize thediscreteorbit
nent spinor we obtain the Dirac equation in
of the point group by functions on the dual
momentum space
space. We formulate the following conditions
(γµp −m cI)ψ(p ) = 0 (4) for these constraints:
µ 0 µ
i) they should vanish on and only on the
We recover the Dirac equation in posi-
points of the orbit
tion space (3) by making the substitution
ii) they should admit a periodic extension
p = 2tan πk ε in (4) and then applying
µ ε µ µ
the discrete Fourier transform iii) they must be Lorentz invariant
1/2ε iv) the difference equations should go to
F(n) = exp (−2πiknε)Fˆ(k)dk (5) the continuous wave equation when the
Z−1/2ε lattice spacing goes to zero.
The difference equation (3) satisfies iv) but REFERENCES
leads to the constraints
4 1. M.Lorente in Symmetry Methods in
(tan πk ε)(tan πkµε)−m2c2 = 0
ε2 µ 0 Physics (N. Sissakian, G.S. Pogosyan ed.)
Dubna, 1996, p. 368-77.
that violate conditions i) and iii), therefore
2. M. Lorente, J. Group Th. Phys. 1 (1993)
the representations characterized by these
105-121.
constraints are reducible.
In the problem of fermion doubling one 3. M. Lorente, Lett. Math. Phys, 13 (1987)
considers the inverse of the propagator, such 229-236; Phys. Lett. B 232 (1989) 345-
as 350.
4 4. M. Lorente in Symmetries in Science IX
(sin πk ε) (sin πk ε)−m2c2
ε2 µ µ 0 (B. Gruber,M. Ramek ed.) Plenum, N.Y.
1997, p. 211-223.
is invariant under the cubic group and be-
come zero on the points of the orbit under 5. M. Lorente, P. Kramer in Symmetries in
this group, but is zero also in the border of Science X (B. Gruber, M. Ramek ed)
the Brillouin zone −1/ε ≤ k ≤ 1/ε. There- Plenum, N.Y. 1998 p. 179-195
µ
fore the representation is not irreducible and
can be reduced giving rise to different ele-
mentary systems with the same mass.