Lecture notes from a series of lectures delivered at the Summer Study Institute on Combinatorial Mathematics and Its Applications in the Natural
Sciences, held on June 2-27, 1969, in Chapel Hill, and sponsored by the
U.S. Air Force Office of Scientific Research under Air Force Grant No.
AFOSR-68-l406 and the U.S. Army Research Office (Durham) under Grant No.
DA-ARO-D-3l-l24-G9l0.
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6 0
8
9
3
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1
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6 5 9 8
1 0 6 9
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9 8 7 6
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3 4 5
4 5 6
5 6 0
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0 1 2
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8
COM BIN A TOR I A L
MAT HEM A TIC S
YEA R
7
February 1969 - June 1970
RELATIVITY GROUPS IN A FINITE SPACE-TIME
by
E.G. Beltrametti *
and
A. Blasi *
Institute of Statistics ~limeo Series No. 600.13
Department of Statistics
Univeristy of North Carolina at Chapel Hill
JUNE 1969
*Istituto
di Fisica Teorica de11'Universita di Genova, Italy
Istituto Nazionale di Fisica Nuc1eare-Sez.di Genova
Relativity Groups in a Finite Space-time
by
E.G. Beltrametti and A. Blasi
Istituto di Fisica Teorica dell'Universita di Genova, Italy
Istituto Nazionale di Fisica Nucleare-Sez.di Genova
INTRODUCTION,
In these lectures, we shall discuss some points
concerning the question:
how far can the physical space-time be thought
x1 ,x 2 'X 3
of as a finite collection of points whose space co-ordinates
and time co-ordinate
k marks?
X
o
take values in some finite set
(k)
containing
To be sure, the question is quite unpopular in physics since
the language of dynamical physical laws is the one of differential equations.
In
~ite
of this, it may be shown that some symmetry properties of
the finite space-time look attractive for physical interpretations.
We
shall be mainly concerned with the symmetries displayed by standard relativity groups, namely three-dimensional rotations, proper and improper
Lorentz transformations acting onto the finite space-time.
that
(k)
1
wv
x,
v
be closed under linear transformations
E
(k)
be a finite field
Letting
C.
1
+
C.
J
w
= Lv
1
wv
x,
v
which leave invariant some quadratic form of the vari-
xO'x 1 ,x 2 ,x 3 ;
abIes
Xl
This imposes
thus we are naturally led to assume the set
GF(k)
ations in the field:
to
[1].
CO' C , .•. , C k 1 be the elements of
1
(addition) and by
(k)
C·C.
1 J
GF(k),
denote by
(multiplication) the two binary oper-
in particular
Lecture notes from a series of lectures delivered at the Summer
Study Institute on Combinatorial Mathematics and Its Applications in the
Natural Sciences, held on June 2-27, 1969, in Chapel Hill, and sponsored
by the U.S. Air Force Office of Scientific Research under Air Force Grant
No. AFOSR-68-l406 and the U.S. Army Research Office (Durham) under Grant
No. DA-ARO-D-3l-l24-G9l0.
2
(i)
a unique element, say
under addition
0;
i + Co
exists having the properties of "zero"
= C
i
o
1;
noted by
C '
1
(C'C
n
being a prime,
pn
V C.):
= C.,
l
will be de-
it will also be de-
1
1 11
-1
C.
an integer:
1
when multiplied by
1
or
1
As well known, a finite field
by the number
C.
exists having the properties of "unity"
the unique element giving
will be denoted by
1
when added to
1
under multiplication
C·
it will also be denoted by
-c. ;
a unique element, say
Ci
GF(k)
exists only if
GF(pn)
moreover
k=pn,
p
is uniquely determined
of its elements, since all finite fields of the same
order are isomorphic.
the sum
V Ci ):
'
the unique element giving
noted by
(ii)
(C
CO'
1+1+ ... +1
The integral mark obtained by iterating
will be denoted by
m.
m
times
Let us recall in particular
that
PX
(0.1)
Le.,
GF(pn)
=
0,
has characteristic
p,
and that Fermat's theorem holds:
= X,
(0.2)
The finite space-time we are concerned with is thus what is usually called a Galois geometry.
From a mathematical point of view, these
geometries have been extensively studied especially by the school of
B. Segre [2].
The problem of whether a Galois geometry can be adopted
for a description of the physical space is not at all straightforward; it
"
has been worked out some years ago by the Finnish school of G. Jarnefelt
and P. Kustaanheimo [3], showing that, under some restrictions on the
order
pn ,
they are actually
nonheretical for physical purposes.
problem will be shortly accounted for in Section 1.
This
3
We shall not try any reformulation of dynamical physical theories
within a Galois geometry: in fact, little has been done
[4];
in this direction
let us only mention that any field theory would be free from usual
divergencies since all integrals would become finite sums.
Beyond this
dynamical aspect of the problem, H.R. Coish [5] and I.S. Shapiro [6]
pointed out the appearence of a number of symmetry properties caused by
the very fact of finiteness of space and which are independent of the
number of points in it, (i.e., independent of the order of the basic coordinate field
GF(pn));
the last author particularly suggested the
relevance of these properties for the fundamental theory of weak interactions of elementary particles.
In the present study, we are moving
along similar lines; part of the results has been already given in previous papers [7,8].
In Section 2, the three-dimensional rotation group
and the proper homogeneous Lorentz group over a Galois geometry are defined and their homomorphisms with
2x2
matrix groups are worked out.
The relevant modular representations are then discussed in Sections 3,4.
Adjoining the space reflection to the proper Lorentz group, we come to
some relevant properties of Dirac spinors over a finite field (Section
5):
in particular we shall treat (Section 6) the properties displayed by
bispinor sesquilinear forms (e.g., the ones occurring in weak-interaction
Hamiltonian).
If the abelian group of space-time translations is also
taken into account one is led to the study of the so called Poincare
group and some problems arise concerning its modular representations:
this is shortly discussed in Section 7.
4
I.
CoMMENTS ON TI-lE PHYS ICALLY ALLOt/ED FIN ITE FIELDS.
In con-
structing a new physical theory on finite fields, one has to bear in
mind a correspondence principle with classical physical schemes.
one needs some minimal algorithm of algebraic operations in
analogous to the usual algebra.
Thus
GF(pn)
It is first of all required to introduce
"positive" and "negative" numbers so that the usual rules of signs hold
true for algebraic operations.
GF(pn)
elements of
X E GF(pn)
For this purpose, let us recall that the
may be divided into "squares" and "not squares":
is said to be a square if
otherwise it is called a not-square:
y
GF(pn)
E
the element
x=y2,
exists such that
a
is not recognized
either as positive nor as negative and the number of squares equals the
one of not squares.
Of course, the property of being a square or a not
square is combined under multiplication according to the sign rule:
in
this respect, squares and not squares may be assumed as positive and
negative elements respectively.
However, in order to have consistency
with the sign rules when transposing monomials from one side of an
equality to the other, we further require the opposite
element
ment
-1
X
E
GF(pn)
to be a not square (and viceversa).
whole field and
GF(p) ;
-1 = 1 ,
wP
GF(pn)
GF(pn)
if and only if
since
we shall have
to be a not-square becomes
ing to
P2 1
w, w2 ••• wp-1
w
= 2k-l
P2 1
= -l.
or
p:: 3
we know, that the not-squares of
n
is odd, while for
n
Then we get the requirement
(1.1)
Thus the ele-
should be a not square, in analogy with real numbers.
be a primitive root in
( -1)
of a square
-X
p - 3
(mod. 4),
n
odd.
GF(p)
even
and
a
Now, let
span the
The condition for
(mod. 4).
Pass-
remain such in
they become squares.
5
As a further step in the analogy with usual algebra, the condition
GF(pn).
In fact,
GF(p2n):
.2 = -1.
in par-
(1.1) allows us to introduce a "complexification" of
GF(pn)
any element of
becomes a square element of
i
ticular there is an element
able to prove that any
x,y
E
GF(pn),
n
=
GF( P2n)
GF(p2n)
E
Z.
GF(pn)
®
,
such that
can be written as
and the complex conjugate
p -th power of
GF(p2n)
Z
E
z* = x-iy
We are now
x+iy,
is obtained as the
Actually, by a dimensionality argument, we get
GF(pn)
and by use of (1.1) it easily follows
n
,.p = - i
(1. 2)
,
hence we get (see (0.2))
n
n
xP +(iy)P
(x+iy)P =
(1. 3)
n
=
x-iy
=
(x+iy) *
p
since the remaining terms in the sum have coefficients divisible by
and vanish because of (0.1).
We may give a meaning
"smaller than"
x-y
«)
to the notion of "greater than"
defining
X
>
Y
or
X
<
Y
square (positive) or not square (negative).
(»
and
according to whether
However, it is impor-
tant to recognize that the sum of two squares can be a not square, so
that there is lack of transitivity for inequalities.
As a consequence,
GF(pn)
would be deprived
a geometry whose points have co-ordinates in
of usual metric relations:
this raises the question of whether a finite
geometry can have anything to do with physical space.
In this respect,
it may be shown [3] that, under conditions like (1.1) the transitive ordering may be ensured over a subset
GF( pn):
a very rough estimate gives
E
of
N
N~ in p.
consecutive elements of
This ordered region has
been called Euclidean, since metrics can formally be introduced in it.
6
It should be borne in mind, however, that
E
is not an algebraic
field.
GF(pn):
Consider, e.g., a three-dimensional space over
subspace of points having co-ordinates in
tice with usual geometric relations.
the
E looks like a finite lat-
By allowing
p
to become suitably
large, this lattice may approximate the observed physical space with
arbitrarily high accuracy; however, we are left with the problem that it
is not closed with respect to algebraic operations of the basic field
~ve
believe that such difficulty is not crucial "a priori":
ac-
cording to Shapiro [6], it cannot be taken for granted that the suggestion to use a space-time manifold devoid of metrical properties is wrong,
for in the microcosmos the classical space-time concepts lose their direct
physical content and exist inasmuch as there exists a theory inherently
consistent and in agreement with experiments.
To avoid notational complications, we shall assume
here on:
P is replaced by
pn.
We quote some results about quadratic equations.
of sets of solutions
GF(pn)
integer,
C·
1
and
d
1
= -1
v
non-zero ele-
is given by
v = p
n
The number
of the equation,
C X2 + C X2 + .... +
2 2
1 1
where
from
it will be clear that the results given in the next sections
hold unchanged if
ments of
n=l
n(2m-1)
- np
n(m-1)
is square
if
II
is not-square.
7
Similarly, for
where
is square
\'1 =
=
2.
-1
is not-square.
II
LORENTZ AND ROTATION PROPER GROUPS OVER
L(4,p)
proper Lorentz group
GF(p).
Define the
as the group of invertible linear sub-
stitutions
3
I
Xl =
(2.1)
II
R..
\1=0
X
llV
•
v'
which leave
invariant,
with elements
R(3,p)
group
= 0.1,2,3;
II
R..
llV
)
•
1.
.{. Tgl).{. = g,
e.,
I)
X , R..
II
(R..
det R..
GF(p);
E
llV
4x4
stands for the
9
=
+1,
matrix
= d i ag (1, - 1, - 1, - 1) .
The sub-
of the proper three dimensional rotations is similarly
formed by the substitutions:
3
X~
(2.2)
1
leaving
GF(p)
j =1
X2+ X2+ X2
1 2 3
with
L(4,p)
(2.3)
= I
r .. x.;
lJ J
invariant.
rOO = 1,
and
= 1,2,3;
i
1
We denote by
r Oi = rio = O.
R(3,p)
X. ,
r ..
lJ
E
r
GF(p);
the
Note that
4x4
rTr = 1.
are finite groups of order
2
det r
~R(3,p) = p(p -1) .
= +1 ,
matrix over
8
Consider now the group
SL(±)(2,p2)
of
u,S,y,o
(2.4)
it is a finite group of order
2~L(4,p)'
formed by the matrices having determinant
sider also the subgroup
SU(±) (2,p2)
E
2x2
matrices over
GF(p 2 );
det a = ±1;
~L(4,p)'
has order
SL(±)(2,p2)
of
SL(+)(2,p2)
while the subgroup
+1
GF(p2);
Con-
formed by the mat-
rices
det u = ±1 ,
(2.5)
det u = aa*+SS* = ±1
where we notice that the condition
may be solved
in a finite field for both choices of the sign, since a sum of square ele-
SU(±)(2,p2)
ments may be a non-square element.
SU(+)(2,p2)
while its unitary subgroup
determinant
+1
has order
has order
2~R(3,p)
formed by the matrices having
The orders of these groups show that
~R(3,p)'
it is impossible to have, according to the simplest analogy with the
classical case, a
and of
to
R(3,p)
1
to
2
homomorphism of
onto
2 homomorphism and by
L(4,p)
onto
Indeed, denoting by
(- -
-:»
a
2
the transition to a subgroup, one
may prove the scheme [5]
I
::/SL(+)(2,p2 L
I
"~SU(+)(2,p2)
SL(±)(2,p2): /
"
>
~SU(±) (2,p2)
I
R( +) (3,p)~
L(4,p)
I
'R(3,p)~
JI....--_ _>~_-I
>
9
SL(±)(2,p2)
Explicitly, the homomorphism between
a, -a
ciates to the pair
(2.6)
where
.e.
fJV
= J;; (det a)
°0 ,° 1 ,° 2 ,° 3
conjugate of
a,
L(4,p),
and
asso-
the Lorentz transformation
Sp(o fJ a
°v
-r
a),
a
E:
are the usual Pauli matrices and
(of course the factor
verse of the "integral mark"
(1+1)
of
J;;
at
is the hermitian
in (2.6) stands for the in-
GF(p).
The remaining homo-
morphisms shown in the previous scheme are particular cases of (2.6).
Let us remark that the elements of
rotations for which
Sp( r)
R(+)(3,p)
is a square in
may be thought of as those
GF(p).
this kind could also be given for the subgroup
A characterization of
L(+) (4,p)
homomorphic to
SL(+)(2,p2).
3. MoDULAR REPRESENTATIONS OF RC3,p) AND LC4,p)
Reiner [9], Wigner [10], Hamermesh [12]).
(Curtis and
When facing the representation
problem of the finite geometry groups, we need to choose the field on
which the representations have to be defined.
Besides the usual possi-
bility of building representations over the complex number field
representations),
(C-
another possibility is self suggesting, i.e., of
building representations over a finite field (modular representations).
About modular representations, we should say that the classical definition
works backwards.
Anyway, about their properties more could be said than
what has been written here.
List of classical theorems.
(i)
No. of equivalence classes
= number
of irreducible representations.
(Proof can be given by group algebra.)
10
In our case, we have to consider the formula
Sn
1 (n + ')(n + ~)
=
which is the sum of the squares of the first
applied to
(iii)
Ipl
n
integer numbers, which
does not yield the correct expression.
Theorem on unitary equivalence of representations of finite order
groups, (Wigner) also fails since there is a sum over all representative
elements.
Here we shall be mainly concerned with modular representations: they
may be built up explicitly [7] but some care must be
of classical theorems no longer apply [9].
paid~
for a number
Classical theorems which do
not hold include:
MASHKE THEOREM:
every reducible representation of a finite group is
decompos ab Ie.
This theorem does not hold, since the proof rests on the sum over
all group elements, which yields a multiplicative factor equal to the
order of the group; but which
by
a (mod
-
p)
if their order is divisible
p.
PROOF:
We need to find a projection
F
which decomposes the vector
space into a direct sum of invariant subgroups and this projection must
commute with all the group representatives.
F = -'nG EXEG T(x)ET(x)-l
visible by
p.
E,
we set
which does the trick as long as
In particular,
ducible modular representation
which commutes with
For
0(9),
V9
Schur's lemma now read:
O(g)
E
G,
of a group
G,
rl
G
is not di-
given an irre-
the only matrix
is a multiple of the unit matrix.
This is a necessary (but not sufficient) condition for a modular representation to be irreducible.
11
A few words on the distinction between p-regular and p-singular
elements:
Example, why shall we need only p-regular classes
trix with elements in
~ [~l ~2)
=>
(A)
1
Ai
with
GF(p).
and
k = (A) k = 1
2
ma-
If it happens that we can diagonalise it
A2 e
GF(p),
which at most can be
V
while if the matrix has period
np
k
then if the matrix has period
p-l
since
xeGF(p),
then
and the matrix can not be represented in the field.
The classical theorem linking equivalence classes to irreducible
representations becomes:
characteristic
p;
if
let
K
G
be a finite group and
is a splitting field for
of irreducible, inequivalent representations of
the number of p-regular equivalence classes of
is a splitting field for
K
presentation of
An element
X
visible by
p);
Gover
of
G
K
G
K
G,
a field of
then the number
Gover
K
is equal to
G.
if any irreducible modular re-
K.
remains irreducible for any extension of
is p-regular if
x
k
= 1
with
k
%p
(k
not di-
a p-regular class is formed by p-regular elements.
Note
that if an equivalence class contains a p-regular element, then the whole
class is p-regular since
(yxy
-1
)
k
k -1
= yx y
.
Let us remark that a sufficient condition for
field for
G
is that all the m-th roots of
K
belong to
to be a splitting
K,
is the least cornmon multiple of the orders of the elements of
Consider first the group
R(3,p).
SU(±)(2,p2)
where
m
G.
which is homomorphic to
We shall make use of Weyl's method by introducing as a basis of
12
GF(p 2 ),
the carrier space the homogeneous monomials in
N(j)cj+m j-m
m c,
n
=
where
rn
j+m
j -m
and
: N(j) ' E, , nEG F( p 2) ,
m
are both integers or half integers with
u
U,
-j::;
m ::; j,
j
~
and
O.
is written in the form
U E
under
j
are both positive integer numbers, i. e.,
= [a,S]
=
s]
[a
-S* a*
aa* + SS* = ±l
we set
=
=
which defines a representation of
D~l~ :~) (u)
with
E
GF(p2)
SU(±)(2,p2)
of dimensionality
(2j+l),
given by
(3.1)
mi n(j+m,j-m')
\
j+m) [ j-m) j-ml-k *j+m-k k(_ *)k-m+m'
k+ml-m a
aSS
.
[ k
/
k=max( 0 ,m-m' )
Eq.
(3.1) is just the familiar one [10] but
(see eq.
and
S
are now elements
(2.5)), as well as the normalization factors
(to be specified later).
sentation
a
Moreover, it will be proved that the repre-
is irreducible if and only if
j ::;
.2.:l
2
13
D{j){u)
Clearly, if
is a representation, also
det u
is a representation of
SU{±){2,p2):
it is not equivalent to
and is likewise irreducible if and only if
j
sionality label
S;
P.:.l
2
Thus, the dimen-
no longer suffices to specify the representation:
e,
introducing a new two-valued label
found
j
O{j){U)
all the irreducible representations
so far will be written in the form:
h: 1 , .... ,
J' = 0 '2'
P.:.l
2
(3.2)
e=O,l.
Now we provide the proofs for the above statements about irreducible representations.
PROPOSITION
I.
p-regular and
The group
GF{p2)
SU{±)(2,p2)
has all its equivalence classes
is a splitting field for the group.
We shall consider only the subgroup
tension to the whole group is trivial.
classes individuated by the
ment:
SU{+){2,p)
has
since the exp
equivalence
vaZues of the trace of the generaZ ele-
then each equivalence class contains an element of the form
- (a
-c* ac)
V -
p
SU{+){2,p2),
a
with
E GF{p),
C
E GF{p2)
and
a2
+
cc*
nonsingular matrix
1
c
Ja 2-1
(tEGF{p2))
T = t
c*
2
Ja -1
1
= 1.
Now, the
14
TvT- 1
is such that
of
V.
Then, if
[L+
=
o
vk = 1
0]
+
2
L-=a
,a-1
±
where
L-
are the eigenvalues
(TVT- 1 )k = Tv kT- 1 = 1
it follows
k is either equal to
of its divisors, and in any case
k
t
p,
which implies
V
and so
2
P -1
or one
is p-regular.
Hence all equivalence classes contain a p-regular element and are
therefore p-regular.
G
of the elements of
roots of
for
1
GF{p2).
and we know all
This proves
Similarly it is proved that
SU{±){2,p2)
representations over
classes of
2
P -1,
is at most
belong to
SU{+){2,p2).
field for
Moreover, the least common multiple of the orders
GF{p2)
is a splitting field
GF{p2)
is a splitting
so that the number of irreducible, inequivalent
GF{p2)
SU{±){2,p2),
is equal to the number of equivalence
namely
2p.
Moreover, we get:
PROPOS ITION
2
I
The representations
e = 0,1
O{j,e){u), O <J"<cl
-- 2 '
are all the irreducible, inequivalent modular representations of
Let us restrict to the subgroup
O{j,O){u)
SU{+){2,p2)
and to
O{j){u)
since the extension to the general case is obvious.
of the irreducibility is by induction on the index
j.
The proof
is irre-
decible since it is one dimensional; suppose the same is true for
and assume
the existence of a
is reducib le.
2{f+1) + 1
A{u)
C{u)
=
j = f
The reducibiZity assumption insures
square matrix
V
such that
15
A(u)
where
B(u)
and
are irreducible representations according to the
A(u)
induction hypothesis; furthermore
B(u)
is equivalent to
O(i-S+~)(u)
O(s)(u)
is equivalent to
for some index
S
i+l.
<
and
It fol-
lows:
Tr(O(i+l)(u)) = Tr(A(u)) + Tr(B(u)) = Tr(O(S)(u)) + Tr(O(i-s+~)(u)).
The above relation, applied to the particular element
u
=
(g
a~J,
yields
i+l
i-s+~
S
'\L a s-r a *s+r + /\
r=- s
-'-----
\
/
m=-i-l
t=-i+s-~
aa*
Notice that the l.h.s. polynomial contains powers of
dif-
ferent from those of the r.h.s. and the equality cannot be satisfied with
a
a
E
GF(p2)
*2(i+l)-p
~
p, in which
a*2(i+l) = a2(i+l)-p so
unless
and
2(i+l)
case
a
2(i+l)
We have thus proved that
O(j,O)(u)
2(i+l)-p+p =
a
that the role of
is interchanged and the degree of the polynomial in
p.
=
aa*
a*
is lowered by
' <
J -
are irreducible for
but we still have to show they are certainly reducible for
and
j
>
Q:l
2
18
2
This is simply shown by referring to the Weyl basis of monomials
f~j)(~,n) = N~j)~j+mnj-m;
m
>
and
or
P21
and m
m = .2.2 + m'
<
if
j
>
P2 1
the monomials corresponding to
. t sub space:
- P-21 form an'lnvarlan
then
infact set
j = Q + jl
2
16
Thus, under a transformation of the group
i.e., a linear combination of monomials of the same type.
This concludes
the proofs of irreducibility for modular representations.
Through the homomorphism, Eqs. (3.1), (3.2) determine the corresponding representations of
(_1) 2j O(j,O;e)(u)
tations of
R(3,p) ;
remarking that
it is seen that for integer
R(3,p)
representations of
j's
the true represen-
are obtained, while for half-integer
R(3,p)
j's
one gets
up to a sign ambiguity (as in the classical
case) .
By a procedure similar to the one used for
SU(±)(2,p2),
find out the irreducible modular representations of
SL(±)(2,p2);
one may
they
have the explicit form
, 1 , ... ,.P.::..l
2
where
and
of
o(j)(a)
is obtained from (3.1) replacing
respectively (see eq. (2.4)).
y
SL(±)(2,p2)
is
2p2;
a*
and
(-S*)
by
6
The number of equivalence classes
therefore all the irreducible representations
are exhausted by (3.3) since the previous Proposition 1 is easily extended to
SL(±)(2,p2).
(See, Brauer and Nesbitt, Ann. Math.
~,
556
(1941) .)
It is easily seen that
(O(k)(u))*
is equivalent to
O(k)(u):
from (3.3) it then follows that all the inequivalent irreducible repre-
17
SU(±)(2,p2)
sentations of
O(j,O;e)(u),
may be written as
according
to the notation used in (3.2).
In the following, we shall be mainly concerned with the spinor representations, namely
j+k
=~,
and it is worthwhile to remark that
(3.4)
4.
UNITARY RAY-REPRESENTATIONS,
Unitarity:
UU+
property, applied only to matrices such that
about matrices.
U+U
sketched very briefly:
for
j
1 and nothing
The proof by Wigner may be
it amounts to setting
A = ZXEG O(X) . O+(x),
and diagonalize it, take square roots and so on.
A = 0
=
As remarked in previous notes, the theorem about uni-
tarity fails for modular representations.
A = A+
=
Here we mean the
In our case,
half integer.
Consider now the unitarity character of the modular representations
of
R(3 ,p.
)
( 2j
t!
Choosing
so that
D(j,O;e)(u)
integer
j~
group~
in (3.1) such that
~s unitary for integer j~
but it ~s not for half-
contrary to the classical case.
At first
the
(j)
Nm
sight~
one could guess
that~
owing to the finite order of
the non unitary representations could be made unitary by a
suitable equivalence transformation; this is not the case since the classical
theorem ensuring this possibility is no longer valid for modular represen-
18
tations;
~n fact~
the proof of the theorem fails if one works on a field
in which a sum of positive elements may be non-positive [12].
Indeed,
one may easily prove that such an equivalence relation does not exist in
our case [7].
Q,M, AND
RAy
REPRESENTATIONS:
A symmetry group in q.m. is in-
tended as a group which leaves invariant the physically observable values,
i.e., the measurements.
Such measurments are expressed in terms of prob-
ability of a certain quantity to attain a given value in the experiment
at hand.
probability that it will end up in state
¢
¢
to a physical state?
of our group which acts on
H
{e
i8
¢}
H,
the
I(¢,~)
H is given by
E
Then, for what measurements may one associate a ray
a vector
~ E
For instance, suppose that system is in a state
I.
rather than
In terms of rays, the representation
need no longer be a vector representation
but rather a ray representation, i.e., a representation as a phase factor.
In general, we might try to keep vectors associated to states in q.m. and
employ ray representations for the symmetry group.
(The question arises
whether a ray representation is equivalent to a vector representation.)
DEF,
RAy
REPRESENTATION,
D(u)D(u ' ) = w(u,u')D(uu ' ),
case we want it unitary, we must have
w(u,u
l
)
Iw(u,u
l
)
I = 1.
which in the
Furthermore,
must satisfy other relations which come from associativity of
multiplication.
Anyway, changing from
D(u) ~ D'(u) = ei¢D(u)
will not
change matters at all for what concerns a ray representation, (Hamermesh
[12]).
However, the unitarity of the
R(3,p)
modular representations may
be recovered by letting them become ray-representations [12] (actually,
to describe the transformation laws of physical states in quantum mechanics, one only needs ray-representations).
To show this, let us
19
introduce, instead of (3.2)
det u,
=
(4.2)
where we remark that the equation for
choices
det u = +1
and
det u = -1
Xu
has
{14].
p+1
solutions for both
The unitarity relation
(4.3)
D(j,O;e)(u)
may now be deduced from (4.1), but
plication law up to a phase factor.
D(~,O;e)
spinor representation
obeys the group multi-
To be more explicit, consider the
= (det u)e
• u
(see eq. (3.4)) and the
corresponding
= det u,
(4.4)
its multiplication rule reads
(4.5)
D(~,O;e)(u) . D(~,O;e)(u')
and it follows
W ,U
u
D(~,O;e) (uu I),
=
det u det u'
I
•
w~, U I = '::":;d:"::e""'t-=-u--='U=-T-'''''':::''-
= 1,
so that all properties of a ray-representation are met
by the factor system
{wU,ul}.
The unitary class of equivalence associated to the unitary rayrepresentation
D(u) is formed by all representation
any solution of
V being any unitary matrix and
cu . c*u = 1 '
i. e."
what we call a "phase factor".
We may thus state that all the ray representations deduced from
(4.4), by changing the choice of
XU'
belong to the same unitary class
20
of equivalence:
and, clearly
in fact the choice
X~ [X~] * __
would correspond to
1.
Xu Xu
Similarly, the two ray representations
D(~,O;l)(u)
D(~'O;O)(u)
and
belong to the same unitary class of equivalence, since
D(~,O;l)(u) = (det U)D(~'O;O)(u)
i.e., a phase factor.
det u
and
is either
+1
-1,
or
As a representative of this equivalence class,
let us assume
(4.6)
=
being a fixed solution of x X *
uu
=
det u.
To meet a classical analogy, we now look for representations of
SL(±)(2,p2) which become unitary when restricted to SU(±)(2,p2):
this
is done by a generalization of the vector representations (3.3), into
ray-representations according to the procedure previously discussed.
place of
In
(given by (3.4», we introduce
(4.7)
the unitarity property (4.3) is met by restriction to
SU(±)(2,p2)
and
the multiplication rule
(4.8)
D(~,O;e)(a)
. D(~,O;e)(a')
=
wa,a D(~,O;e)(aa')
l
exhibits the properties of a ray-representation.
,
By allowing
X
a
to
assume different solutions in (4.7), one obtains ray-representations of
the same equivalence class; the same happens going from
e=O
to
one may thus assume, as a representative of the equivalence class
e=l;
21
(4.9)
=
\vhere the label
a* = det a.
X
e
has been omitted, and
X
a
is a fixed solution of
Similar remarks hold for the case
k=~;
j=O,
X
a
by the same
procedure, we individuate an equivalence class whose representative He
choose to be
=
(4.10)
D(~,O)(a)
and
X a*.
a
D(O'~)(a)
become equivalent only when restriced to
SU(±)(2,p2).
PROPER1Y OF GAUGE TRANSFORrvlATIONS AND MJLTIVALUED REPRESENTATION,
A multivalued representation, where with the matrix
=
p+1
is associated
(j = ~)
the set of representatives
which associates
U
det u,
representatives to a group element thus enlarging
the rotation group to that of
2x2 GF(p2)
matrices with
Idet\ =1
and
thus it coincides with the extended orthogonal group defined by Coish.
According to his reasoning, one could then define a gauge transformation
which could possibly be associated with change.
The identification of
a
E
+
2
SU-(2,p
)}
with the extended orthogonal group of Coish might be carried out as
follows
i)
solutions of
XX*
=
k
= 1 , 2 , ... ,p +1 ,
Ie
L,
-_
Qp-1
I-'
22
ii)
rr*
solutions of
-1
= ~
V- a
E
=
k+~
k
=
1,2, ••. ,p+1,
~ = p
p-l
we set
c+
C::::> C,
=
{~ka
¥
C-
=
{t;;k+ a
C
=
{n:
=
1
=>
det( n)
V- a
E
V- n
=
t;;j
E
C
=
c+
u
C
SL(-)}
det( n) det(n) *
conversely since
Idet(n) I
SL(+)}
= 1}
C
j even => t;;j/2 sti 11 solution and hence
det(n/;j/2) = 1
j odd
and
=> ~j/2
n
=
sati sfi es t;;j/2 • (t;;j/2) *
i ~j/2b
b E SL(-)
=
-1
See Coish [5] for what should be done about change.
5. DIRAC SPINOR REPRESENTATION.
dimensional representation (the group
Lorentz group
L(4,p)
The possibility of having a two
SL(±)(2,p2)
itself) of the proper
is related to the possibility of summarizing Eqs.
(2.1) and (2.6) in the alternative form
~
= I].l
0"
X ,
].l].l
Let us now consider the extended Lorentz group
by adjoining the space reflection operation
L(4,p),
obtained
P to the proper group L(4,p).
It is easily seen that, as in the classical case, it is impossible to have
23
L{4,p)
a two dimensional representation of
P,
forming with
E(A)*
X
E-1 ,
=
X
E =
L{4,p)
To have a linear representation of
4x4
In fact, trans-
X.
we get the non linear relation
A -+P A{P)
X
(5.2)
A
con
we need to adopt a
basis, e.g.,
(5.3)
on which
P
induces the transformation
A
(5.4)
~]
A{P)
A
X
X -+P XI
=
0
t
=
A
YO xYo'
Let us write the transformation induced by
A
(5.5)
YO
L{4,p)
[~ :] .
= YOt =
in the form
1\
L{4,p)
X ---0>
=
S{a) XS{a)t ,
A
where the hermitian conjugation ensures that
in
GF{p).
striction of
XI
individuate coordinates
According to the discussion of the previous section, the re-
S{a)
is required to be unitary:
to
looking
at Eqs. (5.1), (5.2), (5.3) and taking into account Eqs. (4.9), (4.10),
one is thus led to write
D{~,O){a)
(5.6)
S{a)
=
=
0
where, up to now,
Sa
a
0
e E D{O'~){a) E- 1
a
0
Xa
0
1
a E a* E-
S
is restricted, by (5.5), to be any solution of
ea s*a
=
1
'
24
S(a)
However, we have still to impose that
whole:
be a ray representation as a
this corresponds to the physical requirement of having no more
arbitrariness than an overall phase factor in the definition of the four-
S(a).
component spinor which transforms according to
Taking into ac-
count Eqs. (4.8) and (5.6), it is easily proved that this requirement
implies
=
i.e., the factors
tation of
8
8
aa
I
,
a must provide an one-dimensional vector represen-
SL(±)(2,p2).
According to (3.3), there are thus the two
choices
8
a = 1'
8a
= det a'.
from (5.6), we then deduce the two ray-representations (we make use of
the identity
Ea*E- 1 = (det a)a t
-1)
S(e)(a) =
(5.7)
The corresponding
constructed:
4x4
e=O,l.
spinor representations of
(YO'
they will be denoted by
representative of the space-reflection
course
(YO'
S(O)(a))
and
(YO'
P,
s(l)(a))
L(4,p)
S(e)(a)),
may now be
where
YO
according to (5.4).
is the
Of
are ray-representations (we do
not need to write the explicit form of the factor systems which characterize their multiplication rules):
they are inequivalent.
it is important to emphasize that
Therefore, two kind of Dirac spinors are possible:
the one transforming under
(YO'
and the one transforming under
S(O)(a))
(YO'
which will be denoted as
S(l)(a))
which be denoted as
25
This doubling is peculiar of the finite version
L(4,p)
of the Lorentz
group; its root lies in the appearence of both signs in the determinant
of the matrices forming
SL(±)(2,p2),
the group homorphic to
L(4,p).
Let us finally remark that the problem of spinor representations
of the finite Lorentz group has been treated also by I.S. Shapiro [5],
although he is concerned only in two-dimensional spinors giving up the
possibility of representing explicitly the operation of space reflection.
6, PROPERTIES OF BI-SPINOR FORMS,
~(e),
peculiarities of the Dirac spinors
relevance.
Consider two spinors
precisely:
sesquilinear) forms
We shall now discuss some
which could be of physical
~(e), ¢(e)
and their bilinear (more
(6.1)
B(,)
being some
4x4
matrix whose tensor nature is specified by a set
of indices shortly denoted by
(,).
Such quantities, often called
"currents" in physical literature, are the one appearing, e.g., in
electromagnetic and weak-interaction Hamiltonians.
the question of whether a choice of
with respect to
L(4,p).
B(T)
We shall now raise
exists making (6.1) covariant
As a guide, let us recall that in the classical
continuous case the analogous question leads to the construction of five
covariant currents, i.e., a scalar and a pseudo-scalar a vector and an
axial vector, a second order tensor.
We omit at once the currents
e 'f e l
,
26
L(4,p),
for they cannot be made covariant with respect to
as it could
be deduced from the following analysis.
The transformation law of the current (6.1) under the space reflection
P
1j;(e)T B
<j>(e)
(T)
(6.2)
t
YO
note
P
1j;(e)
>
t
is given by
• t • B
• Y • <j>(e)
YO
(T)
0
'
YO
1j; (efr B(T) <j>(e)
(6.3)
L(4,p)
and the proper subgroup
L(4,p)
>
t
1j;(e) • S(e)(a)t • B • S(e)(a). <j>(e).
(T)
To answer the previous question, one has thus to check whether
YO·B(T)·YO
S(e)(a)t.B(T)·S(a)
and
the transformation laws, under
P
B(T)
are related to
and
L(4,p)
respectively, which
characterize
a scalar:
(S)
a pseudo scalar:
rYO BYO
(PS)
i
=
-B
ls(e)(a) B S(e)(a)
=
B;
a vector:
YO B~ YO
=
_(-1)
<5
O~
~
(V)
S(e)(a)t B S(e)(a)
~
B,
=
~
= 0,1,2,3
t
3
1 (det a) I SP(er aer a ) Bv
2
v=O
according to
~
v
27
-
an axial-vector:
= (-1)
6
Ofl
B
lJ
(A)
()
5 e (a)
t
()
B 5 e (a)
v=O
fl
-
= ~1 (det a) I3 5p(o ao aT). B ;
fl
v
v
nd
a 2 -order tensor:
= (_ 1)
+6
6
Ofl
Ov B
,
flV
fl,V
= 0,1,2,3
(T)
To solve these equations, it is useful to introduce the notation
(6.4)
and correspondingly (see (5.4), (5.7))
(6.5)
(det a) atb(T) (l)a
(det a)e atb(T) (2)a t
-1
(det a)e a-1 b(T) (3)a
(det a)a- 1 b(T) (4)a t
-1
Of course, the homogeneous equations
be satisfied by the
B(T) 's
(5), (P5), (V), (A), (T)
identically with respect to
a
E
have to
5L(±)(2,p2);
28
the label
e
appears as a parameter and only for
formally reproduce the classical continuous case.
e=O
these equations
A straightforward
calculation leads to the following results
(A)
r
r
if
(PS)
,
e = 0:
B = Y
0
e = 1:
no solution;
e = 0:
B = YoY '
s
if e = 1:
Ys = i
(- 1 0
0 1) ;
no solution;
o '
[ -Ot]
0
(V)
for both
e = 0,1 :
B
(A)
for both
e = 0,1:
Bj.l = YoYsYj.l
(T)
r
e = 0:
if e = 1:
= YoYj.l'
j.l
Yf =
of
f = 1,2,3;
= YOYj.lY
B
j.lv
v
no solution
The relevant feature of these results is the very special role
played by the vector and axial-vector currents, the ones of paramount
importance in elementary particle physics.
Referring to the construction of hamiltonians which occur in weak
interaction theory, the problem arises of multiplying two currents to
obtain a scalar or a pseudo-scalar quantity.
According to our previous analysis, if at least one of these currents is built up with spinors of the kind
e=l,
then necessarily only
vector and axial-vector currents can be used in such hamiltonians, according to the empirical evidence.
It could be tempting to associate the
29
e=l:
leptons to the choice
the
VtA
currents appearing in weak inter-
actions would then be the unique possibility in the framework of a finite
geometry.
The
Y matrices we have defined obey the relations
1
YlJv + YVlJ
=
2g
lJV
lJt V =
t
O,lt 2 t3
9 =
a
a
a a
0
0
-1
a
a
0
-1
0
0
0
-1
=
thus they provide a realization of Dirac matrices (which coincides, up to
a factor
(-i),
with Weyl's representation).
case t the sixteen matrices
In analogy to the classical
YO' YOYst YOYlJt YOYSY ' YOYlJY v
lJ
(lJ
<
v),
are
a complete set and this ensures that the currents we have examined are
the only irreducible ones.
y's
Of course t the particular realization of the
4x4 basis
we have found is related to the particular choice of the
made in (5.3).
7.
POINCARE
Lorentz group
P(4,p)
GROUP OVER GF(p).
The Poincare or inhomogeneous
is the group of linear transformations
det t
x,a
lJ
lJ
E
= 1;
GF(p).
30
Its elements are denoted by
l
l
(a,£)(a ,£') = (a + £a , fll)
(a,£)
with composition law
a
stands for the 4-vector
where
- 1 a, £ -1) .
( ao' a , a , a ) • ( a,£ ) -1 = (-£
3
1
2
The subgroup of translations
has a semidirect product structure.
P6(p4_ 1 ),
p2(p4_ 1 )
i.e.,
of
1
(a l ,£I)-1(a,l)(a ' ,£') = (£1-
invariant since
P(4,p)
T(4,p) = {(a,l), Va}
a,l)
T(4,p);
The order of
p4
the product of the order
E
T(4)
,p
of
is Abelian
thus
P(4, p)
is
with the order
L(4,p)
We shall now briefly recall the procedure (due to Wigner) to construct the irreducible representations of the usual Poincare group; this
classical sketch will clarify the physical meaning of the numbers associated with irreducible representations.
First one finds the irreducible representations of the translation
subgroup; since this is abelian, they are one dimensional and individuated
by a character
Ix q (a) I = 1,
(a,q)
xq(a)
hence
xq(a)
-+ -+
= aoqo - a q.
= e-i(a,q)
(q)
contains also every character
orthochronous Lorentz group
G
q
where
= xq (a 1
+ a 2 ),
= (qO,q1,q2,q3)
q
and
These characters are then classified into orbits or
equivalence classes; the orbit
"little group"
x (a ) x (a )
q 1
q 2
which satisfies
c
L+
X
Aq
+
Lt.
which contains the character
where
A
belongs to the proper
To every orbit
(q)
one associates a
which leaves invariant a point of the orbit
(different points on the same orbit have isomorphic little groups).
There are four different orbits individuated by the 4-vectors
a)
(m,O,O,O),
b)
(m,m,O,O),
with corresponding little groups
c)
R(3,p),
(O,m,O,O),
E2 , L2 +1 , L:
d)
(0,0,0,0)
31
Each orbit is characterized by a "quantum nUlllber" which is the
length of the representative vector; it is clear that only the orbits of
type
a)
and
b)
are of direct physical interest, for the associated
m
"quantum numbers" are
and
0
and they individuate particles of mass
m or zero mass (e.g., mesons and photons respectively).
type
c)
The orbit of
corresponds to an imaginary mass and that of type
d)
to a
particle with null four-momentum.
If then, one knows the irreducible representations of the little
groups, it is possible to reconstruct the representations of the whole
group.
The little groups provide a second "quantum number", which in
case a) is the spin
j
~
0 integer or half-integer, and in case b) is
the helicity, integer of half-integer also.
In principle, the same rea-
soning can be applied to modular representations, but care must be paid
when treating the translation subgroup
T(4,p).
About abelian groups, we have [9]:
Given an abelian group
finite field
K
H
of characteristic
of order
p,
if
h
and exponent
pis, H
GF(p)
even if
T(4,p)
alence classes but one are p-singular.
(a,l)n
we have
(0,1)
is
p
since its order is
were a splitting field for
if
=
T(4,p)
all
In fact, chosen
h
K.
4
P ,
moreover,
T(4,p)
(a,l)
(na,l)
so that the least integer for which
a f O.
This implies that
T(4,p)
and a
has less than
distinct irreducible one dimensional representations over
The theorem applies to
S
equivE
T(4,p)
(na,l)
=
has just one irre-
ducible modular representation, i.e., the identical one.
Forced, as we
are, to abandon the idea of strictly modular irreducible representations
of
T(4,p)
we may proceed along two different lines in order to recover
results of physical interest:
32
(i)
extend the base field to a larger one, which is essentially the
field obtained adjoining to
n
=
GF(p)
the complex numbers
e2nin/p
0, 1 , ... ,p- 1.
(ii)
give up the requirement of irreducibility and substitute it with
that of indecomposability for the representations of
T(4,p)
The first alternative leads back to a classification completely
analogous to the classical one, the representatives of the translations
q = (QO,Ql,q2,q3)
are labelled by 4-vectors
X (a )
Q
= exp ( 2ni~ga)),
and
(a,l)
corresponds to
where
=
(Qa)
To these representations one may apply the theory of induced representations with results completely analogous to the previous given ones, the
specturm of the momenta
q
is obviously contained in
GF(p).
This pro-
cedure has the serious drawback that we now don't exactly know on what
type of space the matrices operate, due to the extension of the base
field.
The second alternative has a branching point with two possibilities:
I)
Theorem (Curtis & Reiner).
pS
indecomposable inequivalent representations over
A cyclic group of order
pS
has exactly
GF(p) which have
1,2, ... , pS.
dimension
Since
T(4,p)
is the direct product of
4
T(l,p)
need only find the indecomposable representations of
groups, we
T(l,p)
given by
1,
[~l
2
1 a a
0 1 2a ,
a a 1
3
1 a a2 a
a 1 2a 3a 2
a a 1
0 a a
3a
which are
33
and so forth, repeating the preceeding matrix, adding a last row of
zeros except the last element which is one and completing the last col-
(a+l)n,
umn with the expansion of
matrix itself.
Obviously, we have
n being the dimensionality of the
p
of them since
aP
=
1
V a
E
GF(p).
To these representations, we cannot apply the theory of induced
representations as sketched here, even though we might obtain from each
one of them a representation of the whole group
P(4,p).
This procedure,
on the other hand, does not insure to obtain indecomposable representations
of
P(4,p)
II)
T(4,p)
nor that we have a sufficient number of them.
We can employ 2-dimensional indecomposable representations of
labelled by a 4-vector
(a, 1)
q
~
= (QO,Ql,Q2,q3)
(6
and defined through
(a Q)]
1
•
To these, we can apply the theory of induced representations, regaining
all the results already shown in (i) and with the bonus that we do not
have to extend the base field.
to make, for we have
p4
This which looks like a possible choice
inequivalent indecomposable representations
which possibly give rise to indecomposable representations through the
little groups of
P(4,p),
suffers from need of theorems.
We have not
been able to prove neither completeness nor indecomposability for the
induced representations of
investigations.
P(4,p)
and the subject is still open to
34
REFERENCES.
[1]
For a general treatment of Galois field theory, see, e.g.,
B. Segre: "Lezioni di geometria moderna", Vol. 1 (Bologna,
1948); L. Dickson: "Linear groups" (New York, 1958).
[2]
See, e.g., B. Segre: "Introduction to Galois geometries", Atti
Accad. Naz. Lincei Mem. C1. Sc. Fis. Mat. Natur. !, 135
(1967).
[3]
"
G. Jarnefe1t:
Ann. Acad. Sci. Fennicae A. I., No. 96 (1951);
P. Kustaanheimo: Pub1. Astron. Obs. Helsinki, No. 32 (1949);
No. 34 (1952); No. 52 (1957); No. 55 (1957).
[4]
H. Joos: Journ. Math. Phys. 2, 155 (1964); Y. Ahmavaara: Journ.
Math. Phys. ~, 87, 220 (1965); 1, 197, 201 (1966).
[5]
H. R. Coish: Phys. Rev. 114, 383 (1959).
[6]
1. S. Shapiro: Nucl. Phys. 21,474 (1960).
[7]
E. G. Be1trametti and A. A. Blasi: Journ. Mat. Phys.
(1968) .
[8]
E. G. Be1trametti and A. A. Blasi: Nuovo Cimento 55A, 301,
(1968); Atti Acc. Naz. Lincei, C1. Sc. Fis. Mat. Natur. 44,
384 (1968).
[9]
For a general treatment of modular representations, see C. W.
Curtis and 1. Reiner: "Representation theory of finite groups
and associative algebras" (New York, 1962), Chap. 12.
[10]
See, e.g., E. P. Wigner: "Group theory and its applications to
the quantum mechanics of atomic spectra" (New York, 1959),
formula (15.21).
[11]
See, e. g. , the book of ref. [10] , Chap. 9, p. 74, 75.
[12]
See, e.g., M. Hamermesh: "Group Theory", Chap. 12 (Reading, Mass. ,
1962).
[13]
See Dickson's book quoted in ref. [1], p. 47, 48.
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