Finite Dimensional Represntations

of CCR by pxp Matricies Over Finite Galois Fields (Wikipedia)
and of FCCR over Finite Galois Fields
FCCR Table of Contents

A reference site for some of the following is ABSTRACT ALGEBRA ON LINE: Contents

Begin with a highly condensed review of conventions, definitions and notations and theorems from abstract algebra without proofs:

Some standard definitions of group (Wikipedia) and subgroup are assumed.

A module M is an additive commutative group; the additive identity is denoted by 0. A submodule of M is a subgroup. A ring R is a module with an added multiplication operation (not necessarily commutative) where both right handed and left handed distributivity as well as associativity hold. If a ring does not have an identity, it can always be appended. Any ring R with unity is isomorphic to the ring of all its right or left multiplications. Every ring is isomorphic to a ring of endomorphisms of some commutative group. A subring of R is subset of R closed under the operations of R. This is a stronger requirement than being a submodule. A submodule A of R is a subring iff AA is contained in A. A domain is a commutative ring with a unity element. A commutative ring is a ring with commutative multiplication.

   The cancellation law in a domain D:
        With  a,b,c in D
        if c not= 0 and ca = cb
        then a = b.

   An element b of a ring R is said to be
   a divisor of zero if,
        With  b,c in R
        if b not= 0, c not= 0 and bc = 0.
        then a = b.

An integral domain R is a domain with cancellation law. The cancellation law is equivalent to the nonexistence of divisors of zero. Although J_p, p a prime is an integral domain, J_n, with n not prime is a domain but not an integral domain. In J_n, n not prime, all elements do not have inverses. Some elements do; there is a commutative multiplicative group formed by the elements that do have inverses, and the order that group is the value of the Euler function phi(n), which equals the number of integers less than n and greater than zero that are relatively prime to n. Clearly, for any prime p, phi(p) = (p-1).

   An ordered domain D is a domain is
        a domain with a subset of positive elements D^+ congruent to
        D^+ + D^+ is properly contained in D^+
        D^+ * D^+ is properly contained in D^+
   with the
        Law of Trichotomy:
        a in D  => (a = 0) or (a is in D^+) or (-a is in D^+)

The ordering of a domain D implies the cancellation law, and therefore that D is an integral domain. The countably infinite domain of all integers J, is an integral domain. Similarly the Gaussian integers J[i] is an integral domain.

A division ring R is a ring wherein the elements in R^* the nonzero elements of R, all have multiplicative inverses.

The elements of a ring generate cyclic submodules M_lambda, by replicating themselves under addition. Let Card( M_lambda ) denote the order of a submodule.

          c( R )  =  sup Card( M_lambda ) < infinity

the characteristic of R is c( R ). If every nonzero M_lambda has unbounded order, R is said to have characteristic 0. Otherwise, there is at least one module M_lambda with finite order. For any such module there will also exist a module in R whose order exceeds that of M_lambda, and in this case R is said to have characteristic infinity. The characteristic of a subring of R does not exceed c( R ).

If A and B are submodules of a ring R, the Submodule Product AB is the set of all finite sums

             SIGMA  a_lambda b_lambda
             a_lambda in A, b_lambda in B

A submodule A of a ring R is a Right Ideal iff

             RA  is contained in  A
   a Left Ideal iff
             AR  is contained in  A
   and an Ideal (Two-sided ideal) if A is both a
   right sided ideal and left sided ideal.

The zero element is clearly always an ideal in any ring, and called the Zero Ideal. Any ring is an ideal in itself and is called the Unit Ideal. A ring which have only the two above Trivial Ideals, is called a Simple Ring. (A division algebra is a simple ring.) Semisimple Rings are direct sums of simple rings. Ideals of whatever type are closed under sum and intersection. Every element r of a ring R determines an ideal called a Principle Ideal denoted (r). If b and c are elements of a ring R, A is an ideal in R,

             b  =  c + A

   determines an equivalence class of elements so related,
   denoted {b} and called a residue class of R modulo A.

The intersection of any two (right, left, two sided) ideals is a (right, left, two sided) ideal. The product of two (right, left, two sided) ideals is an ideal contained in their intersection. Ideals are of interest because in algebraic geometry, they are abstract descriptions of algebraic varieties in rings of polynomials and because they are also the kernels of ring homomorphisms.

A Group Homomorphism is a mapping from a group G to a group G' which preserves the group operation. A Ring Homomorphism is a mapping from a ring R to a ring R' which preserves the ring operations. The image of a homomorphism is never "larger" that its preimage. In a ring, the preimage of zero is the kernel of the homomorphism. If the kernel is zero, the homomorphism is an isomorphism.

A field is a set F equipped with two binary operations, '+' (addition) and '*' (multiplication). F is a commutative group with respect to '+' and the additive identity is denoted by '0'. Let F^* denote the elements of F with with 0 deleted. F^* is also a commutative group with respect to '*' and the multiplicative identity is denoted by '1'. The two operations together further satisfy the distributive law,

        For any a,b,c in F
        a(b + c)  =  ab + ac

where notation is simplified by denoting the '*' operation by adjunction. The order of a field is its cardinality as a set.

A field may also be casually defined as commutative division ring. The characteristic of a division ring is either 0 or a prime. The characteristic of a field is its characteristic as a division algebra. Any finite integral domain is a field.

An Algebra is a ring with scalar multiplication, the scalars being elements of a field; in short, "a ring over a field". A Vector Space is a module over a field. An R-Module is a module over a ring.

The primitive finite fields J_p, of characteristic and order p are frequently introduced by the classic procedure of constructing the "field of quotients" of two integral domains. In this construction, the field elements (or marks in the language of [Dickson 1900] ) are residue classes of the integers J modulo the prime p. In this representation scheme, every element of J falls into one of the residue (equivalence) classes, with the equivalence of two integers being given by,

             k (mod p)  =  j (mod p)
   Then, e.g.,
             -1 (mod 3)  =  2 (mod 3)

so -1 and 3 belong to the same equivalence class, and could each represent the same element of J_3. Such a representation allows that one can write down any element of J that is equivalent to one of the positive integers

             0, 1, 2, ..., (p-1)

and J_n is an integral subdomain of J.

Consider a slightly more abstract approach where the field is considered as simply a set of symbols with the field axioms satisfied in terms of a multiplication table and an addition table. The field elements can, however, be uniquely and suggestively symbolized by the integer symbols designated above. This is to say that the symbols of J_p will always be represented by positive integers. We will include the concept of "inverse" of addition by attribution of an operation rather than marking the elements, it then the binary operation that has an inverse rather than an element with respect to addition. The third operation of subtraction of two positive integers can still always be defined in J_p as the inverse of the operation of addition. The J_p have an implicit toroidal topology T^1, having as consequence that they are not ordered fields. The elements may be arranged on a circle. As convention say that passing from 0 to 1, 1 to 2, and penultimately p-2 to p-1 flows in a counterclockwise direction, the usual positive direction convention for angles in a plane. The final counterclockwise step (p-1) to 0, closes the circle. To perform the operation of addition (k+j) graphically, for k and j in J_p, start at k on the circle and take j integral steps in the counterclockwise direction. Read off the symbol associated with the position. To perform the operation of subtraction (k-j) graphically, for k and j in J_p, start at k on the circle and take j integral steps in the clockwise direction. Read off the symbol associated with the position. At no time is a concept of "negative number" needed, nor is it introduced.

The finite fields J_p contain no proper subfields and are therefore prime fields. Every prime field is either the rationals or a J_p. The J_p do not, of course, exhaust the category of finite fields. Every field is an extension of a prime field. Finite fields may also be defined as root fields, also called splitting fields associated with the solutions of polynomial equations with coefficients in J_p, and, as such, as extensions of J_p. A root field inherits the characteristic of its prime field.

Let x be some indeterminate entity that admits a formal multiplicative replication, so that exponents of x can be defined

             x E  =  E x  =  x
             x^0  =  E
             x^n x^m  =  x^(m+n)

where n and m are arbitrary integers, and E is a multiplcative identity for x. The powers of x form a countably infinite, commutative, multiplicative group. Define polynomials in x as finite linear combinations of powers of x with positive exponents, and coefficients in some ring R. The set of polynomials in x over R denoted R[x] is itself a ring which contains R. The degree of a polynomial is equal to its highest power. For infinite linear combinations (formal power series), the notation R[[x]] is usually used. If D is a domain (commutative ring), then D[x], is also a domain which contains D. If D is an integral domain, then D[x] is also an integral domain which contains D.

Let D be any integral domain. It turns out that D may always be embedded in a field Q, which is the field of quotients of D. [Think of the embedding of the integers in the field of rationals.] Also, from an integral domain one may construct the field of quotients by considering the ordered pairs

       Q = {a/b: a, b in D and b not= 0}
   where multiplication and addition are defined by
       (a/b)(c/d)  =  (ac/bd)
       (a/b) + (c/d)  =  (ad + bc)/bd
   and the embedding of D in Q is given by the map
        a  ->  (a/1)

Since any finite integral domain is a field F, its field of quotients is isomorphic to F itself.

Let D[x] be the space of polynomials over a finite integral domain; it is also then a space of polynomials over a field. Multiplying a polynomial p(x) by a nonzero member of D does not change the roots of the polynomial. There is an extension of D, the root field of the polynomial that contains all the roots of p(x), so it is meaningful to talk of the roots of p(x) and establish a partition of D[x] into equivalence classes that are characterized by the sets of roots. A polynomial with coefficient 1 for for its highest term, then respresents an equivalence class. A Separable Polynomial over F is one whose roots lying in its root field are all distinct. If that polynomial is the characteristic polynomial of a matrix operator, the operator is said to have zero spectral mutiplicity. A polynomial that is not separable is surprising called an Inseparable Polynomial over F.

   For any polynomial f(x) over and arbitrary field,

        f(x)  =  a_n x^n + a_(n-1) x^(n-1)

                 + a_(n-2) x^(n-2) + ... + a_0

   define the formal derivative

        f'(x)  =  n a_n x^(n-1) + (n-1) a_(n-2) x^(n-1)

                  + (n-2) a_(n-2) x^(n-3) + ... + a_1

where a natural multiple of any element of the field is easily and clearly defined. If f(x) is separable, then f(x) and f'(x) are relatively prime, and f'(x) not= 0. Said differently, f(x) and f'(x) have no roots in common. Then also, as formula (provable by relatively simple induction) for the formal derivative,

   Let r_k be the distinct roots of f(x), then

        f(x)   =  PI (x - r_k)
        f'(x)  =  SIGMA f(x)/(x - r_k)

Let R[x] denote the ring of polynomials in the indeterminate x, with coefficients in R. R[x] is an integral domain iff R is an integral domain. The set of polynomials denoted J_p[x] satisfies the axioms of an integral domain.

   For f(x), an element of J_p[x], if
   there exists an element x of J_p such that

             f(x)  =  0
             x is said to be algebraic over, J_p,
             x is said to be transcendental over, J_p,

For any polynomial f(x) over any field F, there is a root field in which the polynomial may be written as a product of monic irreducible polynomials

        f(x)  =  PI (x - a_k)

   with a_k  in  F'

The category of fields that consists of all J_p and their extensions are called Galois Fields GF[p^n]. It is a classic theorem that any field of finite order is a Galois field. Another related theorem is that two finite fields of the same order are necessarily isomorphic. A root field GF[p^n] can be considered a vector space of of dimension n over J_p the prime subfield of GF[p^n]. The number n is called the degree of GF[p^n]. The prime subfield of a finite field is the submodule of the field generated by unity. A primitive root of a field is an element that generates all field elements by powers of itself. Every Galois field has a primitive root. If f(x) is a polynomial with coefficients in a field F over which x is transcendental ( f(x) is irreducible ), and F' is root field extension over F within which the equation

               f(x)  =  0

has solutions ( f(x) is reducible ). F is a subfield of F', and the group of automorphisms of F' leaving F invariant is called the Galois Group of F' over F or the Galois Group of f(x).

We happen to be interested in fields containing certain square roots, and consider as examples, the details for the Galois fields of lower orders, distinguishing the even prime p=2 and the odd primes p > 2.

For the Galois fields GF[2^n], every element is a square. Every element has one and only one square root.

For p > 2, GF[p^n], there are equal numbers of nonzero elements that are squares and notsquares. In the generation of GF[p^n] by a primitive root, the even powers of the root are squares, the odd powers are not squares. The product of two squares and the product of two notsquares are squares, while the product of a square and a not sqaure is a notsquare. [Dickson 1900], p.44.

Consider the matricies B(2) and B!(2) defined for n=2 and equation (2.1), where the matrix elements are taken over the binary field J_2. Performing the binary arithmetic to calculate the commutator, one obtains,

                 [B(2), B!(2)]  =  I(2)
                 {B(2), B!(2)}  =  I(2)
                 N(2)  =  B!(2) B(2),

This is the first and almost trivial example of a representation of CCR in terms of creation and annihilation operators in finite dimensions. If one wishes to pass to the Q-P representation, the field must be extended. We will do this in a minimal way.

To addition and multiplication tables for J_2 are written respectively as

           +  0  1              *  0  1
           -|------             -|------
           0| 0  1              0| 0  0
           1| 1  0              1| 0  1

   To the two elements add a third called 'j', so that the extended Field
   elements can be written as a vector space over J_2,

             a + bj

   where a and b are elements of J_2.

The unique addition and multiplication (there is only one group structure of order 3, and it is commutative) tables of the four field elements of GF[2^2] are then,

           +  0    1   j   (1+j)       *  0   1   j  (1+j)
           -|-------------------       -|------------------
           0|  0   1   j   (1+j)       0| 0   0   0    0
           1|  1   0 (1+j)   j         1| 0   1   j  (1+j)
           j|  j (1+j) 0     1         j| 0   j (1+j)  1
       (1+j)|(1+j) j   1     0     (1+j)| 0 (1+j) 1    j

   From these tables, one sees immediately that every element
   is its own additive inverse,

        a + a  =  0
   and that
          j^(-1)  =  (1+j)
   and that every element satisfies the equation
        a^4  =  a
   For any element of the prime subfield 
        a^2  =  a
        a + a = 0

Furthermore, the subtraction operation is identical to the addition operation and there is an involutive map of GF[2^2] given by

        j  ->  j^(-1)

   under which

        a + bj  ->  (a+b) + bj

        (a+b) + bj  ->  (a+b+b) + bj  =  a + bj

        (a + bj)(a+b + bj)  =  a(a+b) + b(a+b)j + abj + b^2j^2
                            =  a + ab + (ab + b + ab)j + b(1+j)
                            =  a + ab + bj + b + bj
                            =  a + ab + b

   is a map from GF[2^2] ->  J_2

   Define the Q(2) and P(2) operators by

             Q(2)  =  B(2) + B!(2)

             P(2)  =  j^(-1) B(2) + j B!(2)

Define "Hermitean Conjugation" by matrix transpose composed with the field conjugation of GF[2^2] as defined above.

[Note: the idea of conjugation in a field extension modeled on the operation of "complex conjugation" in the complex field as an extension of the Real field becomes reasonably complicated, and is part of the Galois theory of roots of polynomials. For the story, Field theory and polynomials is a good place to look. Dave Rusin's encyclopaedic pages on mathematics Index via Mathematics Subject Classification (MSC) is generally a very good place to go ferreting for enlightenment in matters mathematical.]

Both Q(2) and P(2) are "Hermitean",

              Q(2) P(2) - P(2) Q(2)  =  I(2)
              Q^2(2)  =  I(2), P^2(2)  =  I(2)
              Q^2(2) + P^2(2)  =  0

remembering that subtraction in the field is identical with addition and therefore also,

              Q(2) P(2) + P(2) Q(2)  =  I(2)
              Q^2(2) - P^2(2)  =  0

Once can enlarge the field from GF[2^2] to GF[2^3], but his does not effect the above commutation and anticommutation relations. Let the elements of GF[2^3] be represented as a three dimensional vector space over J_2, introducing the new basis element k, by

             a + bj + ck, with a,b,c in J_2

The field GF[2^3] has the addition and multiplication tables:

        +   0       1     j    (1+j)     k     (1+k)   (k+j)  (k+j+1)
        0|  0       1     j    (1+j)     k     (1+k)   (k+j)  (k+j+1)
        1|  1       0   (1+j)    j     (k+1)     k    (k+j+1)  (k+j)
        j|  j     (1+j)   0      1     (k+j)  (k+j+1)    k     (1+k)
    (1+j)|(1+j)     j     1      0    (k+j+1)  (k+j)   (1+k)     k
        k|  k     (1+k) (k+j) (k+j+1)    0       1       j     (1+j)
    (1+k)|(1+k)     k  (k+j+1) (k+j)     1       0     (1+j)     1
    (k+j)|(k+j)  (k+j+1)  k    (1+k)     j     (1+j)     0       1
  (k+j+1)|(k+j+1) (k+j) (1+k)    k     (1+j)     j       1       0

        *   0    1     j     (1+j)    k    (1+k)  (k+j)  (k+j+1)
        0|  0    0     0       0      0      0      0       0
        1|  0    1     j     (1+j  )  k    (1+k)  (k+j)  (k+j+1)
        j|  0    j   (1+j)     1    (k+j) (1+k+j) (1+k)     k
    (1+j)|  0  (1+j)   1       j   (1+k+j) (k+j)    k     (1+k)
        k|  0    k   (k+j)  (1+k+j) (1+k)    1    (1+j)     j
    (1+k)|  0  (1+k) (1+k+j) (k+j)    1      k      j     (1+j)
    (k+j)|  0  (k+j) (1+k)     k    (1+j)    j   (1+k+j)    1
  (k+j+1)|  0 (1+k+j)  k     (1+k)    j    (1+j)    1     (k+j)

With the introduction of the new element k that defines the extension from GF[2^2] to GF[2^3] it is clear that while the above Q(2) acquires no new possibilities, a new P(2) matrix can be defined that uses k in place of j. Moreover, the new P(2) will have the same commutation and anticommutation relations with Q(2) as the old one, and the old and new P matricies are linearly independent. This process of acquiring a new basis element in the field extension from GF[2^n] to GF[2^(n+1)] provides a new P operator which is linearly independent of all the preceding P matricies.

If now, k is used as an index or label for the (n-1) basis elements introduced to define GF[2^n], one can write a collection of CCR-CAR representations in the 2x2 matric algebra over GF[2^n] which all have a common Q(2).

   Define the Q(2) and P_k(2) operators by

             Q(2)  =  B(2) + B!(2)
             P_k(2)  =  k^(-1) B(2) + k B!(2)

Define "Hermitean Conjugation" by matrix transpose composed with the field conjugation of GF[2^2] as defined above extended uniformly to all the additional basis elements. All Q(2) and P_k(2) are "Hermitean",

              Q(2) P_k(2) - P_k(2) Q(2)  =  I(2)
              Q^2(2)  =  I(2), P_k^2(2)  =  I(2)
              Q^2(2) + P_k^2(2)  =  0

remembering that subtraction in the field is identical with addition and therefore also,

              Q(2) P_k(2) + P_k(2) Q(2)  =  I(2)
              Q^2(2) - P_k^2(2)  =  0

   Additionally for basis labels k and j,

              P_j(2) P_k(2) - P_k(2) P_j(2)  =  0

So for only one Q(2) there are as many linearly independent and mutually commuting P_k(2) as one chooses to have by choosing the appropriate GF[2^n].

Suppose we enlarge GF[2] to GF[3], and begin over, still using 2x2 matricies. The addition and multiplication tables for GF[3] are:

           +  0  1  2           *  0  1  2
           -|---------          -|---------
           0| 0  1  2           0| 0  0  0
           1| 1  2  0           1| 0  1  2
           2| 2  0  1           2| 0  2  1

The addition table shows a cyclic group, and the multiplication table for nonzero elements shows the unique group structure of a group of order two. The nonzero element 1 has two square roots 1 and 2, and 2 = (-1) has no square root. If the B(2) and B!(2) matricies are understood over J_3, CCR cannot be written, but of course CAR can. Since CAR can be written one might suspect that the SU(2) Lie algebra relations might be written also. For this the field will have to be extended.

   By introducing the square root

             i  :=  sqrt(2) = sqrt(-1)

extend the field to GF[3^2], the field of order 9, representable as a two dimensional vector space over GF[3].

        a + bi is in GF[3^2], with a, b in GF[3].
        i^2  =  2  =  -1

But this is not finished because 2 seems to have only one square root, and it needs to have two. So introduce instead both square roots

        (+|-i)^2  =  2

   where it would appear that both

        a + bi is in GF[3^2],
        a - bi is in GF[3^2],

   must be considered.

If both positive and negative elements of the field are allowed, it appears that the field order 9 must be doubled to 18, but then this cannot be a Galois field since there is no Galois field of order 18. It is apparently forced that the identification

             -1  =  2
              i  =  sqrt(-1)

be made in calculations, and the field remains of order 9. The '-' is absorbed into the operation of subtraction that is now different from addition. The addition and multiplication tables for the elements of the field will not involve negative numbers.

     +    0      1      2      i     2i    (1+i)  (1+2i)  (2+i)  (2+2i)
     0|   0      1      2      i     2i    (1+i)  (1+2i)  (2+i)  (2+2i)
     1|   1      2      0    (1+i)  (1+2i) (2+i)  (2+2i)    i     2i 
     2|   2      0      1    (2+i)  (2+2i)   i     2i     (1+i)  (1+2i)
     i|   i    (1+i)  (2+i)   2i      0    (1+2i)   1     (2+2i)   2
    2i|  2i    (1+2i) (2+2i)   0      i      1    (1+i)     2    (2+i) 
 (1+i)| (1+i)  (2+i)    i    (1+2i)   1    (2+2i)   2      2i      0
(1+2i)| (1+2i) (2+2i)  2i      1    (1+i)    2    (2+i)     0      i
 (2+i)| (2+i)    i    (1+i)  (2+2i)   2     2i      0     (1+2i)   1
(2+2i)| (2+2i)  2i    (1+2i)   2    (2+i)    0      i       1    (1+i)

     *    0      1      2      i     2i    (1+i)  (1+2i)  (2+i)  (2+2i)
     0|   0      0      0      0      0      0      0       0      0
     1|   0      1      2      i     2i    (1+i)  (1+2i)  (2+i)  (2+2i)
     2|   0      2      1     2i      i    (2+2i) (2+i)   (1+2i) (1+i)
     i|   0      i     2i      2      1    (i+2)  (1+i)   (2+2i) (1+2i)  
    2i|   0     2i      i      1      2    (1+2i) (2+2i)  (1+i)  (2+i)
 (1+i)|   0    (1+i)  (2+2i) (2+i)  (1+2i)  2i      2       1      i
(1+2i)|   0    (1+2i) (2+i)  (1+i)  (2+2i)   2      i      2i      1
 (2+i)|   0    (2+i)  (1+2i) (2+2i) (1+i)    1     2i       i      2
(2+2i)|   0    (2+2i) (1+i)  (1+2i) (2+i)    i      1       2     2i

Showing that the elements 1, 2, i, 2i and only these, all have two square roots.

If the three matricies alpha_a are defined

              |0   1|           |0  2i|           |1   0|
      a_1  =  |     |   a_2  =  |     |   a_3  =  |     |
              |1   0|           |i   0|           |0   2|

   then, with the matrix elements understood over GF[3^2],
   the alpha_a satisfy

             [alpha_a, alpha_b]  =  2i epsilon_(abc)  alpha_c

   a representation of the Lie algebra su(2, GF[3^2]).

For 3x3 matrices, however the B(3) and B!(3) can written, and with the elements understood over GF[3^2], they satisfy

                 [B(3), B!(3)]  =  I(3)
                 N(3)  =  B!(3) B(3),

No further field extension is necessary to define the Q(3), P(3) matricies in the usual way from the B(3) and B!(3).

             Q(3)  =  (1/sqrt(2))( B!(3) + B(3) )

             P(3)  =  (i/sqrt(2))( B!(3) - B(3) )

   and these satisfy

             [Q(3), P(3)]  =  [B(3), B!(3)]  =  i I(3)

   The Tables for GF[5] are

           +  0  1  2  3  4     *  0  1  2  3  4
           -|---------------    -|---------------
           0| 0  1  2  3  4     0| 0  0  0  0  0
           1| 1  2  3  4  0     1| 0  1  2  3  4
           2| 2  3  4  0  1     2| 0  2  4  1  3
           3| 3  4  0  1  2     3| 0  3  1  4  2
           4| 4  0  1  2  3     4| 0  4  3  2  1

Again the additive group is the cyclic group of 5 elements, and the order 4 multiplicative group of nonzero elements is not isomorphic to the Klein 4-group but isomorphic to the cyclic group of order four, frequently represented by complex units {1,i,-1,-i}, the isomorphism being given by

        1 -> 1, 2 -> i, 3 -> -i, 4 -> -1

The Klein 4-group and the cyclic group are the only possible structures for a group of order 4. It is a theorem that for any field of finite order, the multiplicative group of nonzero elements is cyclic. Generally then, while the prime fields have and implicit T^1 topology, the general GF[p^n] has an implicit T^2 topology; where p and n can be taken as measures of the major and minor radii, respectively. Only the elements 4 and 1 have square roots, and one needs square roots for 2 and 3 in order to construct B(5) and B!(5). Formally adjoin the two roots, to construct the field GF[5^3] of 125 elements represented as a 3-dimensional vector space over GF[5],

             a + b sqrt(2) + c sqrt(3)

The large tables for GF[5^3] are omitted. They can, however, be given implicitly and more economically through the vector space representation. The purpose of the basic tables is to be able to compute the sums and products of two arbitrary field elements. This is equally well done by providing the additive and multiplicative results for two arbitrary vector space elements. Introduce the two formal square roots of both 2 and 3: +|-sqrt(2), and +|-sqrt(3). Absorb the signs into the subtraction operation as before, so it is understood that

        a + b(+|-sqrt(2)) + c(+|-sqrt(3))  =  a +|- bsqrt(2) +|- csqrt(3)

the two sign alternatives being taken independently. Then also,

             (+|-sqrt(2))(+|-sqrt(3))  =  1
             (-|+sqrt(2))(+|-sqrt(3))  =  4

Then, the addition and multiplication tables are implicit in vector space addition and the addition table of GF[5], and

          (a + b sqrt(2) + c sqrt(3)) (a' + b' sqrt(2) + c' sqrt(3))  =

               (aa' + 2bb' + cc' + bc' + cb')
             + (ab' + ba') sqrt(2)
             + (ac' + ca') sqrt(3)

   where a, b, c, a', b', c' are all elements of GF[5].


Let F be a field. Let the extension of F12, the first order complete quadratic extension of F be the smallest field that contains F as a subfield and that also contains as elements, all the square roots of the elements of F. In F12, all quadratic equations with coefficients in F are solvable.

In any GF[p^n], p > 2, half the elements will have two square roots (i.e. will be squares) and half will not. (the zero element is not counted. There are (p-1)/2 elements which have (and have not) square roots. The root extension field is of order p^(n(p-1)/2), characteristic p, and degree n(p-1)/2 over GF[p]. Since all finite fields of given order are isomorphic one can write within isomorphism

             For p > 3
             F  =  GF[p^n]
             F12  =  GF[p^(n(p-1)/2)]

   Now, we have the following theorems.

    <Theorem J1>:

   For for p=2 there exists an irreducible representation
   of CCR as creation and annihilation operators of dimension p,
   over GF[p^n], for any n > 0.

   For any p > 2, there exists an irreducible representation
   of CCR as creation and annihilation operators
   by p dimensional matrices over GF[p^((p-1)/2)],
   the first order complete quadratic extension of J_p.


   Either at this point obvious at this point, or left as an exercise,
   according to taste.


   p =  3: GF[3^2]
   p =  5: GF[5^2]
   p =  7: GF[7^3]
   p = 11: GF[7^5]

    <Theorem J2>:

   For for p=2 there exist irreducible representations
   of CCR as Q(2) and P(2) operators of dimension p,
   over GF[p^n], for any n > 1.

   For any p > 2, there exists an irreducible representation
   of CCR as Q(p) and P(p) operators by p dimensional matrices over
   the first order complete quadratic extension of GF[p]:
   GF[p^((p-1)/2)], if p^((p-1)/2) = 4m + 1,
   where m > 0 is integral, and
   GF[p^((p+1)/2)], if p^((p-1)/2) ≠ 4m + 1.


   If p^((p-1)/2) has the form 4m + 1, where m is integral,
   the multiplicative (cyclic) group of the nonzero elements of the field
   has a cyclic subgroup or order 4 that is naturally isomorphic to
   In this case the imaginary unit i=sqrt(-1) is not to be added
   since it is isomorphically already there.


A short list of the lowest order Galois fields with the property that the field has order of the form 4m + 1, follows. These are all the possible Galois fields with order less than 100.

        Field                    Order

        GF[ 5   ]                  5  = 4(1) + 1
        GF[ 3^2 ]                  9  = 4(2) + 1
        GF[ 13  ]                 13  = 4(3) + 1
        GF[ 17  ]                 17  = 4(4) + 1
        GF[ 5^2 ]                 25  = 4(6) + 1
        GF[ 29  ]                 29  = 4(7) + 1
        GF[ 37  ]                 37  = 4(9) + 1
        GF[ 41  ]                 41  = 4(10) + 1
        GF[ 7^2 ]                 49  = 4(12) + 1
        GF[ 61  ]                 61  = 4(15) + 1
        GF[ 73  ]                 73  = 4(18) + 1
        GF[ 3^4 ]                 81  = 4(20) + 1
        GF[ 89  ]                 89  = 4(22) + 1
        GF[ 97  ]                 97  = 4(24) + 1
        GF[ 101 ]                101  = 4(25) + 1

    <Theorem J3>:

   For for p=2 there exist irreducible representations
   of CCR as Q(2) and P(2) operators of dimension p,
   with diagonalizing transformation for the canonical
   cyclic matricies in the canonical N(2) eigenbasis,
   over GF[p^n], for any n > 2.

   For any p > 2, there exists an irreducible representation
   of CCR as Q(p) and P(p) operators by p dimensional matrices,
   with diagonalizing transformation for the canonical
   cyclic matricies in the canonical N(p) eigenbasis,

   over GF[p^((p-1)(p-1)/2)], if p^((p-1)/2) = 4m + 1,
   where m is integral,
   and GF[p^((p-1)(p+1)/2)], if p^((p-1)/2) ≠ 4m + 1.

   (We are still clearly looking at these larger fields as vector
    spaces of the prime fields GF[p].)


   From [Theorem J2], and the fact that the p-th roots of unity,
   not equal to 1, will never be included in the fields of [Theorem J2].
   A root extension is then straightforwardly made from the
   Galois fields distinguished in [Theorem J2].


The order of the field clearly grows quite rapidly with increasing p;

   p = 3:  GF[3^2]  =  9
                      since  3^((3-1)/2)  =  25  =  4(6) + 1
   p = 5:  GF[5^8]  = 390625,
                      since  5^((5-1)/2)  =  25  =  4(6) + 1
   p = 7:  GF[7^24] = 191581231380566409216,
                      since  7^((7-1)/2)  =  1343  ≠  4(m) + 1

    <Theorem J4>:

   In any of the above cases:

             Q(p) P(p) - P(p) Q(p)  =  i I(p)

             Q(p) P^k(p) - P^k(p) Q(p)  =  ik P^k-1(p)

   and thus for any polynomial f(x) over the accompanying Galois
   Field, that,

             Q(p) f( P(p) ) - f( P(p) ) Q(p)  =  i f'( P(p) )

   where the apostrophe designates the formal derivative
   as defined before.


   Either at this point obvious, or left as an exercise following the
   procedure of the result in a standard proof in the context
   of QM.


   <Theorem J5>:

   For any representation of CCR over a Galois field with J_p as the
   prime subfield, i.e., as pxp matrices, the number operator N(p)
   is cyclic of order p:

                         N^p(p) = N(p)

   Det[ N(p) ]  =  0, and Tr[ N(p)  =  0.

   And, for any GF[p^n], while Det[ I(p) ] = 1, Tr[ I(p) ] = 0.

   These follow from elementary and standard results of Galois field
   theory.  That Tr[ I(p) ] = 0, in the context of GF[p], is a key
   ingredient for the existence of such reps of CCR, since the trace
   of commutators will also vanish as usual.

   Also notice that the determinants of all Q(p) and P(p) will be
   zero.  Even in FCCR(n), if n is odd these determinants vanish,
   and do not vanish when n is even.


   <Conjecture J0>:

   (Probably true)
   Let K be any field of characteristic 0, and GF
   any finite Galois field, and let n >= 2.
   There are no nontrivial group homomorphisms

             SL( 2, K )  ->  SL( n, GF )


Along this line of making physics discrete by making the number field with which it makes computations finite, [Ahmavaara 1965a], [Ahmavaara 1965b] suggested some large finite field for standard quantum theory.

It should be noted that these results, in no way, contradict the theorems of [Wintner 1947], [Wielandt 1949], and Taussky [Cooke 1950], p. 55 #7, which deny the existence of representations of CCR within any normed algebra. The loophole for genuine finite dimensional IRREPS of CCR is that the algebras of matricies over finite Galois fields are not normable, since the Galois fields themselves are not normable, having, as they do, essentially toroidal topologies induced by that of the underlying primitive Galois Field. A primitive Galois field is not normable due to its lack of well ordering.

An important difference between these representations and the usual representations of CCR involving not only infinite dimensional Hilbert spaces, but also unbounded operators, is that in any finite case, regardless of the field of the algebra, the trace of the fundamental commutator vanishes. This issue is discussed in the context of the complex field in [Section III], where it is shown that an infinite dimensional IRREP of CCR can be understood as a limit of finite dimensional FCCR in the strong operator topology, but definitely not in a norm topology. Convergence in a norm topology is also, obviously, out of question regarding these "proper" representations over finite fields. It may be possible to play with the various limits of field order and matrix dimension; I have not considered that possibility yet.

I would claim that this lack of order of GF[p], hence that of GF[p^n] does not rule out their physical applicability. The number field appearing in a physical theory is a presumed measuring tool for, most particularly and primitively, spatial distances and temporal lengths. From an epistemological standpoint, it is reasonably presumptious to assume in the foundations of a physical model that unbounded or infinitesimal lengths can be measured, even in principle. It should be a matter of principle that they can, in fact, not be measured, since any reasonable conclusion from quantum physics tells us exactly that; so far, in common physical theory, such a principle of impossibility does not exist though it is highly suggested by many investigations concerning the physical meaning of the Planck length, and also for even larger regimes in QED by the limitation set on probing distances with energies exceeding the energy of pair production.

Time, similarly is measured with reference to some standard clock, whose very ruler is already toroidal given the essential, cyclic nature of clocks. To measure infinite time requires either an infinitely sized clock, or an infinite collection of finite clocks. Neither of these do we, nor can we have. Measurements of time on the order of the Planck time or even accurate to a few thousand Planck time units offer similar problems.

In [Coish 1959], a concept of "local ordering" was introduced for finite GF[p] which could be enough from a physical standpoint when the order of the field is made "large enough".

There is a sense of size or absolute value for an element of a finite Galois field, that being its distance from the zero element. This concept should be useful in interpreting probability amplitudes and then probability amplitudes calculated by the usual quantum mechanical rules. By passing to the field of quotients, the probability distributions can be also be normalized.

If one takes the limit of unbounded p, CCR will be satisfied for every p in the sequence; and the associated Galois field will have the same transfinite cardinality as that of the reals, exhibiting a sequence of CCR representations of bounded operators in nonnormalizable operators which has a limit, a representation (irreducible) of CCR of unbounded operators which must be to the standard CCR IRREP of QM in the Number operator (or oscillator representation. Note that the irreducibility of the p dimensional representations has NOT been proved, or even asserted here.

There are many structural questions that need to be answered. Some should be easy, like calculating the determinants of P and Q, generally. Some may not be so easy, as the general question or irreducibility. Matricies (mxm) over and GF[p^n] are clearly a group additively, and the dimension of the additive group is then m²p^n. Within these matricies there is the general linear (multiplicative) group GL(m, GF[p^n]), consisting of those matricies whose determinant is not equal to the zero element of GF[p]. (Det is a group homomorphism from this multiplicative matrix group to GF(p^n].)

Why might such algebraic considerations be of importance to physics besides the suggestive algebraic structure?
At a fundamental planck level, it becomes obvious after enough consideration that much of the conceptual structure upon which physics tends to rely on, e.g., space, time, measures and measurements, discriminations, etc. evaporate into meaninglesness, as does the concept of "observer". What remains as structure and does not become meaningless is an essentially discrete, finite noncommutative algebraic structure.

Consider only only the general linear group over some GF[p^n]. A group algebra exists for this group over any other field; it does not have to be GF[p^n], and we might just as well consider the scalar numbers of the algebra to be complex rationals.

The basic concepts of any Q theory are noncommutativity and probability. Where might the probabilities come from? As noted above, the toridal nature of finite fields makes their interpretation as probabilities difficult since the ordering of elements disappears. There is an alternative, however, that comes from the combinatorics of the group structure which asks in a "metaphysical" way questions of how probable a given structure is for a randomly chosen element of GL(m, GF[p^n])? For example, what is the probability that a randomly chosen element will be cyclic?

Any finite group (by Cayley's first theorem) can be represented as an element of the symmetric group S(n), i.e. a set of permutations of n objects for some n. Similarly, any (continuous) Lie group can be represented by a permutation of the real numbers, which is mappable to the interval [0, 1].

Consider the space of group homomorphisms from GL(m, GF[p^n]) to [0, 1], using combinatorically calculated probabilities as probabilities of existence.

More generally, consider the space of group homomorphisms from GL(m, GF[p^n]) into GL(m, GF[p^n]), again using combinatorically calculated probabilities for types. It should come as no great shock that there is something wrong and simplistic with the CCR of standard quantum theory, nor should it be surprizing that combinatorically determined probabilities appear as essential "at the bottom".

Change the GF[p] field to its complex extension to discuss transition amplitudes from which probabilities can be calculated in the usual way |z|²=p.

The point, of course, is that a fundamental entry into a fundamental finite Q theory though finite noncommutative group theory is not quite as crazy or ridiculous as it might first appear.

It should be noted that for the Heisenberg algebra, there does exist an ordinarily defined 3x3 adjoint representation constructed from its structure constants: [Appendix B]

        | 0  0  1 |      | 0  1  0 |      | 0  0  0 |
        | 0  0  0 |      | 0  0  0 |      | 0  0  1 |
        | 0  0  0 |      | 0  0  0 |      | 0  0  0 |

   which exponentiates with real parameters to the Heisenberg group so
   that a group element can be represented as

        | 1  p  t |
        | 0  1  q |
        | 0  0  1 |

   Regardless of the field, however, this representation of the algebra
   can never be Hermitean, nor that of the group unitary.

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