of CCR by pxp Matricies Over Finite Galois Fields (Wikipedia)

and

and of FCCR over Finite Galois Fields

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.

TheAncancellationlawin 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 adivisorofzeroif, With b,c in R if b not= 0, c not= 0 and bc = 0. then a = b.

AnThe 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.ordereddomainD 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^+)

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.

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

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 lambda with 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 aThe zero element is clearly always an ideal in any ring, and called theLeftIdealiff AR is contained in A and anIdeal(Two-sided ideal) if A is both a right sided ideal and left sided ideal.

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 + acwhere 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

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_1where 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) k and f'(x) = SIGMA f(x)/(x - r_k) kLet 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 then x is said to be algebraic over, J_p, otherwise 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) 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) = 0has solutions ( f(x) is reducible ). F is a subfield of F', and the group of automorphisms of F' leaving F invariant is called the

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) and {B(2), B!(2)} = I(2) with 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 and a + a = 0Furthermore, 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 Then (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) and Q^2(2) = I(2), P^2(2) = I(2) so Q^2(2) + P^2(2) = 0remembering that subtraction in the field is identical with addition and therefore also,

Q(2) P(2) + P(2) Q(2) = I(2) and 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_2The 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) and Q^2(2) = I(2), P_k^2(2) = I(2) so Q^2(2) + P_k^2(2) = 0remembering that subtraction in the field is identical with addition and therefore also,

Q(2) P_k(2) + P_k(2) Q(2) = I(2) and 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) = 0So 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 1The 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]. with i^2 = 2 = -1But 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], and 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 then 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 2iShowing 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) with 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 1Again 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 -> -1The 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)) = 4Then, 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 FDefinition:

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] F_{1}^{2}= 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. Proof: Either at this point obvious at this point, or left as an exercise, according to taste. QED 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. Proof: 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 {+1,+i,-1,-i}. In this case the imaginary unit i=sqrt(-1) is not to be added since it is isomorphically already there. QED

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].) Proof: 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]. QED

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) implies 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. Proof: 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. QED

<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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