```
```
Notes On The Concept of Quantum Set Theory
```
```
```
```

```
```
Introduction

This continues the introductory essay on the physical notions of quantum set theory.

The idea of quantum set theory, while it may sound to be of a mathematical nature, is necesarily of a physical nature if one means to quantize "points" that comprise sets in such a way that they are treated as physical objects with physical properties.

This was actually the idea of the ancients in general matters of geometry. They were thinking about, and talking about physical reality.

Although there is a fair amount of pure mathematics below, the hard part comes when having to decide on the physics of treating the concept so that it becomes reduced to something which adequately deals with "physical space" and its classically approximate locally Euclidean E3.

Though the expression of quantum theory is highly mathematical, the physical theory is indeed physical, hence more specific than any particular mathematics. While the mathematics of set theory, topology and geometry is not constrained by any physical reality, these conceepts can also be taken to the level of physics, where physical constraints are necessary.

Here, we will take the mathematics as constrained by physical reality, and try to express in mathematical terms, still using standard matheamatical logic, how these constraints should be applied to the more general mathematical theory.

A recurring mathematical structure in physics is the Lie algebra su(2), which describes spin, isotopic spin associated with electrical charge, happens to be abstractly identical to the algebra of SO(3) since SU(2) is the universal covering group of SO(3), whose algebra is normally associated with classical and QM angular momentum operators in an E3. For that matter, one might also consider the groups SO(n), Spin(n) associated to an E^n, and their cognate Lie algebras.

SU(2) occurs again in my FCCR(n) algebra, for n=2, as the defining representation of su(2), where it describes a smallest q-oscillator, and would appear to model the smallest and simplest possible quantum object, and therefore be the most likely candidate to model a physical point, or more accurately any smallest physical event (process), and its energetic excitations represented algebraically by the irreducible representations of su(2) which exist for every dimension n > 1.

The su(2) algebra appears later below, but for now, begining at the begining, I construct a genuine algebra of sets, which act together, with a few possibly interesting mathematical remarks, and get to an algebra of Clifford algebra mapping as quantization centering on the power set, outline some known theory of Clifford algebras and their various relationships, and finally get to discussing the nature of any quantum set theory, from the viewpoint of quantum theory and metaphysics.

```

```
Classical Set Theory & Symbolic Logic

I will not belabor the rigorous mathematics in this: that would take up many volumes that have already been written.

There are two common ways of looking at classical set theory, one (symbolic logic) involves the points of some universe of discourse, and a membership relation, while the other first proceeds without a membership relation, but from a set of axioms about some pair of binary relations on a collection of sets.

```
```
Classical Set theory using symmetric difference
and complementation (Kuratowski)
```
```
Classical naïve set theory is most often introduced through the intuitively clear binary operations of union "∪" and intersections "∩". I will assume that these operations are known in their finite applications, as well, the standard interpretive visualizations in terms of Venn diagrams. In particular, the concept of null set; the relations of disjointness and equality are assumed here known and understood.

Interjection: let us take note at this early point that a classical empty set is also a void, a physical space or spacetime "vacuum" is not a void: a vacuum though it may be, it is apparently not devoid of content. This is clear if we understand space or spacetime in the sense quantum field theory. A void cannot properly model a vacuum. Choose your terminology, but distinguish the two concepts.

These two operations alone are not powerful enough to provide definitions for further structure that would match the intuitive ideas of a general naïve set theory.

If one uses additionally, the assumptions of difference and complementation:

Duality of ∩ and ∪, and the dual map "c" of complementation.
```   In addition, the set theoretic difference '-' can be defined by

A - B  :=  A ∩ Bc

where in the context of some specific universal, most comprehensive set
U, one can define "The complementation operator", "c" operating on any
set is

Ac  :=  U - A

```
Boolean Algebras
```
An algebraic axiomatization, courtesy of the structure by George Boole
(1813 - 1864) called Boolean algbera, takes as primitive, the operators
∩, ∪ and -, is given axiomatically:

1.   A ∪ B  =  B ∪ A
2.   A ∩ B  =  B ∩ A
3.   A ∪ (B ∪ C)  =  (A ∪ B) ∪ C
4.   A ∩ (B ∩ C)  =  (A ∩ B) ∩ C
5.   A ∪ 0  =  A
6.   A ∪ (A ∩ B)  =  A
7.   A ∩ (B ∪ C)  =  (A ∩ B) ∪ (A ∩ C)
8.   a ∩ (a ∪ B)  =  A
9.   (A - B) ∪ B  =  A ∪ B
10.   (A - B) ∩ B  =  0

11.   (A ∩ U)  =  A

Then with U posited, an absolute, well defined complementation can
be defined,

U - A  :=  Ac

(A ∩ B)c  =  U - (A ∩ B)
=  U ∩ (A ∩ B)c
=  U ∩ (Ac ∪ Bc)
=  (U ∩ Ac) ∪ (U ∩ Bc)
=  (Ac ∪ Bc)

[Showing the set theoretic principle of duality]

and dually (exchanging ∪ and ∩)

(A ∪ B)c  =  (Ac ∩ Bc)

With
(A - B)  =  (A ∩ Bc)

(A - B)c  =  (A ∩ Bc)c
=  (Ac ∪ B)

Define the "is a subset of" relation 'subs' as a logical equivalence

A subs B  <=>  (A ∪ B  =  B)

```

Although the algebraic characterization of Boolean algebras can clearly be represented as an algebra of sets, that representation isn't required. The objects of the theory can also be represented by logical propositions, in which case the algebra can also be represented by a propositional calculus. Each is in a sense reducible to the other.

However, a Boolean algebra is not at all what one would call an "algebra" in the modern sense of a vector space over a field with a multiplication defined as a binary operation on the vectors. Among other things, there is no field involved. Without a field, and the associated sense of scalar multiplication of vectors, the algebra becomes a ring. In modern terminology, a Boolean algebra in Boole's sense is really a Boolean Ring.

General Rings
```
A ring is a collection R of {A, B, ...} (not necessarily implying even
countability) on which two operations, x and + are defined so that:

1.   R is a commutative group w.r.t. "+", with additive identity "0".
2.   "x" is a binary operation on R
3.   A x (B + C)  =  (A x B) + (A x C)
4.   (B + C) x A  =  (B x A) + (C x A)

A ring is called

associative, if (A x B) x C  =  A x (B x C), and
commutative, if (A x B)  =  (B x A).

Sets will not form a ring under the ∩ and ∪ operations; however -
first define the symmetric difference operator "▲"

A ▲ B  =  (A - B) ∪ (B - A).

where it can be shown that for the symmetric difference one has
(visualize a Venn diagram)

A ▲ B  =  B ▲ A,                           commutativity
(A ▲ B) ▲ C = A ▲ (B ▲ C),                 associativity
A ∩ (B ▲ C)  =  (A ∩ B) ▲ (A ∩ C),         L distributivity

R's distributivity is implied by commutativity.

The ∩ operator is already assumed commutative by axiom 2.

Now, sets with the operations of "▲" and "∩" *do* form a commutative
and associative ring, where "▲" corresponds to "+" and ∩ corresponds
to "x".

It is easy to see that for any A and B, that A ▲ B and A ∩ B are
disjoint:

(A ∩ B) ∩ (A ▲ B)  =  0

Now, assume that there is a universal set U, and reintroduce and
particularize by definition the complementation operator "c",

Ac  :=  U - A

so,

A ▲ U  :=  Ac      defines a unary involutive complement operation

Then, the Boolean ring becomes a "ring with involution", the
complementation operator being the involution (of order 2).

For any A and B,

A ▲ 0  =  0 ▲ A  =  A,       0 is an identity for "▲"
A ▲ A  =  0,       every A is its own inverse under "▲"

A ▲ (A ∩ B)  =  (A - B) redefines the difference operator

=  (A ∩ Bc)

A ▲ B ▲ (A ∩ B)  =  (A ∪ B) redefines the ∪ operator

which is unambiguous by associativity of "▲"

A ▲ Ac  =  U

A ∩ U  =  A      U is an identity for ∩

A ∩ A  =  A      every A is idempotent under ∩

A ∩ Ac  =  0

(A ∩ B)c  =  Ac ∪ Bc

(A - B)c  =  Ac - B

(A ▲ B)c  =  (A - B)c ∩ (B - A)c
=  (Ac - B) ∩ (Bc - A)
=  (Ac ∪ Bc) ∩ (Bc ∪ Ac)
=  ((Ac ∪ Bc) ∩ Bc) ∪ ((Bc ∪ Ac) ∩ Ac)
=  (Bc) ∪ (Ac)
=  (B ∩ A)c

Then,

(A ∩ A) ▲ (A ∩ A)    =  0

(A ∩ A) ▲ (A ∩ A)c  =  U

(A ∩ B) ▲ (B ∩ A)    =  0

[The Clifford relation mimic?]
(A ∩ B) ▲ (B ∩ A)c  =  U

There is a natural binary field structure of {0, U} (thought of perhaps,
as the extrema of an interval or lattice of algebra elements) with
operations "▲" and "∩"

0 ▲ 0  =  0
0 ▲ U  =  U
U ▲ U  =  0

0 ∩ 0  =  0
0 ∩ U  =  0
U ∩ U  =  U

The "∩" operator is the ring "multiplication" operator, for which one
can define a "division" operator ":" by,

A : B  :=  A ∪ Bc
so
A : A  =  A ∪ Ac  =  U  (U is the identity for : )
A : U  =  A ∪ Uc  =  A  (          "             )

U : A  =  U ∪ Ac  =  U  (this does *not* define a specific inverse
function, but rather an "inverse relation")

With this division operation (relation), the inverse of every set is the
identity U.  It is an operation of division that is not really attached
to multiplicative inverses.

The closest, it seems, one can come to a ∩ inverse is A → Ac, but

A : Ac  =  A ∪ A  =  A  consistent with the idempotence under ∩

Ac : A  =  Ac ∪ Ac  =  Ac

(Ac : Bc)  =  Ac ∪ B  inverting "numerator & denominator"
=  B : A

(A : B)c    =   (A ∪ Bc)c
=   (Ac ∩ B) multiplying A inverse by B, and noting
that this is not equivalent to B:A.
=   (B - A)

(A - B)c    =  (A ∩ Bc)c
=  (Ac ∪ B)
=  (B : A)

(Ac : Bc)c  =  (Ac ∪ B)c
=  (A ∩ Bc)
=  (A - B)

The complementation operator relates ":" and "-" to each other the way that
it relates ∩ and ∪.  Some equational relationships (theorems) are
demonstrated:

(A : Bc)    =   (A ∪ B)

(A : Bc)c  =   (A ∪ B)c
=   (Ac ∩ Bc)

(Ac : B)c  =   (Ac ∪ Bc)c
=   (A ∩ B)

A : (B ∩ C)  = A ∪ (B ∩ C)c
= A ∪ (Bc ∪ Cc)
= (A ∪ Bc) ∪ Cc
= (A : B) : C

A : (B ∪ C)  = A ∪ (B ∪ C)c
= A ∪ (Bc ∩ Cc)
= (A ∪ Bc) ∩ Cc
= (A ∩ Bc) ∪ (A ∩ Cc)
= (A ∪ Bc) ∪ (Ac ∪ C)c
= (A : B) ∪ (C ∪ Ac)c
= (A : B) ∪ (C : A)c
= (A : B) : (C : A)

A ∩ (B : C)  = A ∩ (B ∪ Cc)
= (A ∩ B) ∪ (A ∩ Cc)
= (A ∩ B) ∪ (Ac ∪ C)c
= (A ∩ B) : (Ac ∪ C)
= (A ∩ B) : (C : A)

= (Ac ∪ Bc)c ∪ (Ac ∪ C)c
= (Bc ∪ Ac)c ∪ (C ∪ Ac)c
= (Bc : A)c ∪ (C : A)c
= ((Bc : A)c : (C : A))
= ((Ac : B)c : (C : A))

A ∪ (B : C)  = A ∪ (B ∪ Cc)
= (A ∪ B) ∪ Cc
= (A ∪ B) : C
= (A : Bc) : C

(A ▲ B)c    =  (A - B)c ∩ (B - A)c
=  (A ∩ Bc)c ∩ (B ∩ Ac)c

=  (B : A) ∩ (A : B)

```
Measure Algebras

Some of the most interesting topological spaces are metric spaces. [i] In these, there are nice relationships between measures and topology. E.g., the open sets of a metric topology generate a σ-algebra of sets upon which a measure can successfully be defined. This can be played out more formally:

```
A σ-algebra of subsets of a set S is a family F of subsets
such that:
1) The null set, ∅ ∈ F
2) A ∈ F ⇒ S-A ∈ F
3) ∀i∈Z: Ai  ∩ Ai = ∅ for i ≠ j ∧  Ai ∈ F ⇒
∪i Ai ∈ F

```
Measures
```   A real valued measure on S relative to F is a set functional f,

f: F ─→ [0, ∞], such that:
1) f(∅)  =  0
2) ∀i∈Z Ai ∈ F ⇒ f( ∪i Ai )  =   Σi f(Ai)

The second, most important requirement is called countable additivity.

```
Construction from the Hilbert space V
```
There is a constructive argument from very general Q principles that
if S is some finite set of cardinality n, that quantization
of the points starts by assigning to each point a distinct element
of a basis for complex vector space V equipped with a generalized
(possibly indefinite) inner product.  Further, the quantization of
the power set of S is accomplished by mapping the elements of the
basis of V to Γi, the fundamental generators of a Clifford algebra,
CL(V),

{Γk, Γj}  =  Γk Γk + Γj Γk  =  2 (ek, H ej)

where the ek are the basis elements of V, H is the inner product's
ground form, (ek, H ej) is the inner product of ek and ej.

This symmetrized product of the generators Γk then is the image in
the map of the inner product between the basis vectors of V.

This is as good a place as any to note that when the inner product of
a Clifford algebra becomes completely degenerate so that the defining
relation of the generators becomes

{Γk, Γj}  =  0

the algebra so defined is a Grassmann algebra with antisymmetric
"outer" product as the associative product assumed here and represented
by simple notational adjunction, which can be defined for any arbitrary
vector space.

A Clifford algebra then also has a Grassmannian substructure (ignoring
its inner product values, or setting them all to zero) from
which a proper grading can be defined.

CL(V) inherits the complex structure of V, and so is considered as
a complex Clifford algebra.  This has the result that any indefiniteness
of V's inner product is *lost* in lifting to CL(V), since, if

(Γk)²  =  -1, for some k,

we can always replace Γk with (i Γk).  This is an important point
that is easily passed by too quickly: a fundamental contradiction
between Quantum theory and Relativity theory is simply that one
respects a noneuclidean norm while the other demands a Euclidean
norm, so any facile reconciliation between those two formal demands
should not be missed.

One may equally write these last relationships as,

{Γk, Γj}  =  2 δkj

using a Kronecker δ, and where all the Γk can be chosen to be Hermitean.
We will only be speaking of finite dimensional complex Hilbert spaces
here, so no distinction between formal Hermiticy and selfadjointness

```

This complex CL(V) is actually the quantization of a Boolean ring associated with a set S and with its power set 2S. Conceptually, the idea of determinate membership of a point in a set has evaporated, and it has become "fuzzy", but in more complicated a way than classical (simply statistical) fuzziness would imply. Formally, this increased complexity is the result of the admissibility of complex linear combinations of "points", and these having meaning respecting limitations on their distinguishibility.

Moreover, the subsets of the classical S, are (eliminating the linear ordering on points) the precursors of the quantum states of S. This is a generalization of, and in analogy to, the quantum mechanical points in the spectrum of the position operator q (which are perfectly classical!) being associated to the eigenfunctions of q as singleton states. These eigenfunctions are Dirac δ functions, and general states are certain generalized complex linear combinations of δ functions (distributions). Here, however, what is being quantized is not just the points of S as singletons, but the entire subtheoretical structure, i.e., the power set of S.

-----------------
The mapping from a set theoretic ring with ▲, ∩, c to a Clifford algebra with +, adjunction, hodge star. (Or, should it be really the other way around?)
-----------------

```

Fermionic Creation and Annihilation operators from the Γk, defined by

{Γk, Γj}  =  2 δkj

define
ak  :=  (1/sqrt 2) (Γk + i Γk*)
so
ak†  = (1/sqrt 2) (Γk - i Γk*)

Often the Γs are taken to be Hermitean, but most generally, they
don't have to be.  If not, it must also be true that

{Γk†, Γj†}  =  2 δkj

and then

{ak, aj†}  =  (1/2) (2 δkj + {Γk†, Γj†})
-i {Γk, Γj†} +i {Γk†, Γj}
[XYZZY Is this really true?]

{ak, aj†}  =  δkj

-----------------
```
Complex Clifford Algebras and their Representations

There are structural differences in CL(V) depending on whether n is odd or even, and whether the natural ring structure is extended to a real or complex field. While the structure is more complicated for real Clifford algebras, it is fairly simple for complex algberas:

First, every representation of a real or complex Clifford algebra is completely reducible. Let CL(n), be the algebra for a complex V of n dimensions.

For n = 2m, an even n, there is a single faithful, irreducible representation that is isomorphic to the complete complex algebra of 2m x 2m matricies, and so the algebra is simple, and of dimension 2n.

For n odd, n = 2m + 1, there is one irreducible representation of dimension 2m, which is not faithful. A faithful but reducible representation exists; it exists as a direct sum of two irreducible (nonfaithful) representations where the direct summands are not only subalgebras, but also mutually annihilating two-sided ideals in the faithfully representing algebra of dimension 22m = 2n-1. The faithful representation is, of course, not simple, by virtue of it being a direct sum.

So, complex Clifford algebras are faithfully represented by 2m x 2m complex algebras of matricies of complex V's of dimension 2m, and (2m + 1).

The Lower Dimensional CL(V) Algebras, n = 1-6
```
For n = 0,
taken as even, the Clifford algebra (of the "vacuum" or null
set) has dimension 1, and is isomorphic to the complex field
C, which has unitarily equivalent representations as a real
algebra over certain 2x2 matricies, or by extension as a
complex number.  If the underlying Hilbert space is projective,
i.e., is the projective complex plane, the Clifford algebra
is a phase factor, exp( iw ), which is the covering group of
the real line R.

For n = 1,
it is C + C, a diagonal 2x2 complex (reducible) matrix,
for 1 "point Qparticle" = "1 Qpoint particle".  If the
underlying Hilbert space is projective, this matrix
must be Diag[ exp(+iw), exp(-iw)], which is unitarily
equivalent to any element of the Lie group SU(2).

C can also be taken as an irreducible, yet not faithful
representation of CL(1).  For faithfulness, we apparently
need a replication that goes to n = 2.

For n = 2,
The faithful Clifford algebra representation is
expressed by the full (irreducible) algebra of 2x2 complex
matrices M(2).  This also happens to be the complex extension
of the Lie algebra su(2), i.e. sl(2, C) in a direct product
with C.  That is the Lie algebra gl(2, C), structurally
equivalent to the Lie group GL(2, C).  An underlying
projective Hilbert space implies a projective Clifford
algebra isomorphic to sl(2, C), and a lifting of the H(2)
into the C*-algebra of linear operators on H(2) implies
sl(2, C) X sl(2, C), the Lie algebra of the covering group
of the full Lorentz group.

Notice that we have simple built algebras from numbers,
and done nothing explicitly relativistic.  Clifford
algebras are purely about qudratic forms in classical
geometry, so we have also introduced nothing essentially
quantum theoretical, either.
Cf. Classical Geometry & Physics Redux

If n=1 describes a one particle (Qpoint) set, and it needs
an extension to n=2, mathematically, Qpoints must come in
pairs. Think, perhaps, Qpoint-antiQpoint.  This is no more
arcane than the Pauli Principle, and only a slight and
conceptually consistent generalization of it.

If you believe in classical point particles, clap your hands;
if you also believe that local physical space is essentially
Euclidean, clap your hands twice, then two ethereal beings
will be saved from certain painful death at the hands of the
US gummint.  This is, after all, its primary world function.

For n = 3,
the faithful representation is expressed by the direct
(reducible) sum M(2) + M(2) = CL(3), which is a subalgebra
of M(4).

M(2) can also be taken as an irreducible, yet not faithful
representation of CL(3).  The tower of Clifford algebras is

For n = 4,
it is the irreducible M(4). (The Complex Dirac algebra)

For n = 5,
it is M(4) + M(4) which is a subalgebra of M(8) = CL(6).

M(4) can also be taken an irreducible, yet not faithful
representation of CL(5).

For n = 6,
it is the irreducible M(8) = CL(6), etc.

This should be enough to show the pattern of relationships among
the CL(n) elaborated on next in the periodicity theorems.

```
Clifford Subalgebras & The Periodicity Theorems
```
Generally, and in terms of their faithful representations,

CL(2m) is a Clifford subalgebra of CL(2m+1)
and
CL(2m+1) is a Clifford subalgebra of CL(2m+2) = CL(2(m+1))

So the set of algebras {CL(n)} actually forms a nested sequence by the
subalgebra relation, mirroring a nested sequence of classical abstract
sets {Sn} by the subalgebra (subset) relation,

CL(n)  < CL(n+1)

More can be said about the structure of the sequence in n of CL(n).

There is also in Clifford Algebra Theory, the general
"Periodicity Theorem" which says that for REAL algebras for
sufficiently large dimensions, w.r.t. n=8,

CL(n, R)  =  CL(8, R) X CL(n-8, R),  a tensor product

but for COMPLEX algebras, that period is not 8, but 2, so

CL(n + 2)  =  CL(n) X CL(2)

=  CL(n) X M(2)

Formal question:
Is the direct product of any two Clifford algebras a Clifford algebra?
Yes, and

2ⁿ x 2m  =  2(n+m)

So is the direct sum, but only if all the algebras have the same dimension.

2ⁿ + 2ⁿ  =  2ⁿ⁺1

and only if the number of algebras appearing in the direct sum is a power
of two.

Then, one may also form a vector where the basis elements are
Clifford algebras by using these operations together with some
field of scalars.

```
More Formal Clifford Algebra Structure

A basis for CL(n) as a vector space are the k-fold products of the Γk. These monomial forms of degree k classify the geometric antisymmetric tensorial objects that are subspaces of the full tensor algebra over the underlying inner product space:

```
I                scalar            <->     null set
Γi                vector            <->     singleton subsets
Γi Γj                bivector          <->     diploid subsets
Γi Γj Γk        trivector         <->     triploid subsets
...
(n-2) factors        pseudobivector
(n-1) factors        pseudovector
n factors        pseudoscalar      <->     the full subsetset {S}
(n-volume)

Define the symbol CLk(n) as the subspace of CL(n) spanned by the
elements of degree k.  In order to fix the specific bases of the
CLk(n) subspaces, we fix the ordering of the products of Γk
to be a "natural ordering" so that for any monomial

Γi Γj ... Γk

i < j < ... < k

NOTE:
As a formal model of a quantum set theory, the degree of these objects
is the cardinality of an underlying subset in the power set.  It is not
difficult to see that the number of kth degree, linearly independent
monomials is (n k), a binomial coefficient.

Dim( CLk(n) )  =  (n k)

Then, of course, adding the dimensions of all these homogeneous subspaces,
scalar to pseudoscalar (binomial theorem)

n
Σ (n k)  =  2ⁿ  =  Dim( CL(n) )
k=0

as it should.  A state of S associated with definite cardinality k ≤ n
is a complex linear combination of the basis elements that are
homogeneous of degree k.  Such a state then is homogeneous of degree k.

The dual space of linear functionals then is represented by the fully
antisymmetric tensorial objects of V.  The Clifford algebra can then be
found as a subalgebra of the full (free) tensor algebra over V.

More about structure in the dual space later.

```
The Pseudoscalar in Complex Clifford Algebras
```
Let e(n) := e1234...n represent the pseudoscalar (or volume) term of
degree n, then,

│ +1 when n = 0, 1 (mod 4)
e²(n)  =   │
│ -1 when n = 2, 3 (mod 4)

Since we are remaining with complex algebras, we can redefine e(n) by

│ e123 ... n, when n = 0, 1 (mod 4)
e(n) :=  │
│ i e123 ... n, when n = 2, 3 (mod 4)

Then, for all n

e²(n)  =  +1

thus, for a complex CL over any V with signature, that signature
is wiped out in the complex Clifford algebra; not so for a real
Clifford algebra, which makes their structures more complicated.

Now, form the projection operators

P(n+)  =  (1/2)(1 + e(n)),     P(n-)  =  (1/2)(1 - e(n))

so, as required of projection operators,

P²(n+)  =  P(n+),             P²(n-)  =  P(n-)

verifying that these have the property of projection operators.
These project out two nonintersecting ideals of the algebra.

Think of the constructed genuine ring of set theory above under
the operations of symmetric difference and intersection; now, think
of projections projections of any symmetric difference of of sets
A and B onto set A and onto set B.

```
A Uniform Clifford Algebra Basis
```
For k ≠ j,

(Γk Γj)²  =  +(Γk Γj) (Γk Γj)
=  -(Γk Γk) (Γj Γj)
=  - I

requiring one multiplicative interchange.  Similarly, the square of a
trivector monomial will require three interchanges and so square to -I;
e.g.,

(Γk Γj Γl)²  =  +(Γk Γj Γl) (Γk Γj Γl)
=  -(Γk Γj Γk) (Γl Γj Γl)
=  +(Γk Γk Γj) (Γl Γj Γl)
=  -(Γk Γk Γj) (Γj Γl Γl)
=  - I

(Γk Γj Γl Γm)²  =  +(Γk Γj Γl Γm) (Γk Γj Γl Γm)
=  -(Γk Γj Γl Γk) (Γm Γj Γl Γm)
=  +(Γk Γj Γk Γl) (Γm Γj Γl Γm)
=  -(Γk Γj Γk Γl) (Γj Γm Γl Γm)
=  +(Γk Γk Γj Γl) (Γj Γm Γl Γm)
=  -(Γk Γk Γj Γl) (Γj Γl Γm Γm)
=  +(Γk Γk Γj Γl) (Γl Γj Γm Γm)
=  + I

so fourth degree monomials require six interchanges, and so on, following
the pattern for the pseudoscalar.  Let all monomials of degree equal
to 2 and 3 mod 4 acquire a factor of 'i', as is consistent with the
treatment of the pseudoscalar.  Then, if ΓA is defined so that A runs
over all the sequences of indicies that specify all the monomial basis
elements, we have uniformly, for a basis of the full Clifford algebra,

(ΓA)²  =  + I

CONJECTURE:

when
{Γk, Γj}  =  I
then
{Γk*, Γj*}  =  +I  n odd
{Γk*, Γj*}  =  -I  n even

```
The Even Clifford Subalgebra
```
Let CLE(n), the even subalgebra, be the elements of CL(n) for which
the terms of odd degree vanish.  CLE(n) is then a Clifford algebra of
dimension 2n-1.

CLE(n) is, in general not simple: When n is even, e(n) will be a
member of CLE(n); when n is odd it clearly will not.  So then, writing

CLE(n)  =  CLE(n) P(n+) + CLE(n) P(n-)

CLE(n) is seen to fall into two simple ideals when n is even, and
is simple when n is odd.

The map, E(.), the even selector, a Clifford algebra homomorphism,
maps

CL(n)  →  E( CL(n) )  =  CLE(n)  =  CL(n-1)

effectively decrements the dimension of the underlying vector space, V,
by 1, so renaming E(.) as -- (.), define,

-- CL(n)  :=  E( CL(n) )  =  CL(n-1),

while the map,

CL(n)  →  CL(2) X CL(n)  =  CL(n+2)

increments the dimension of V by 2.  Therefore,

CL(n)  →  E( CL(2) X CL(n) )  =  E( CL(n+2) )  =  CL(n+1)

increments the dimension of V by 1, so similarly define the operator
++ (.), writing,

++ CL(n)  :=  E( CL(2) X CL(n) )  =  CL(n+1)

These two operators are like particle annihilation and creation
operators; but what they annihilate and create are Clifford subalgebras.

```
The Clifford Group
```----------

The Clifford Group
is the set of invertible elements in CL(n)
which then exhibits the structure of a multiplicative group.

```
The Discrete Clifford Group
```                is the set of elements in CL(n) that map CL(n) into
the group algebra of its Clifford group.

```
```
Clifford algebras are said, in some literature, to be graded algebras.
Here, however, I'll take the more refined approach of [Chevalley 1955]

```
The Trace Functional Tr(.) & Trace Norm
```
The irreducible representations of Clifford algebras, from which any
finite dimensional representation is constructed are finite dimensional
matrix algebras, and so, a trace operator is well defined immediately
in that no further topological restrictions are required: the properties
of the formal trace on the abstract algebra are valid independently of
any faithful representation, since there is only one such representation.
From the constraint that defines the generators Γk, and the fact that
Tr(AB) = Tr(BA), we see that

Tr( Γk Γj )  =  0, for k not equal to j

│ 2m,  for n = 2m
Tr( (Γk)² )  =   │
│ 2m,  for n = 2m + 1

All the ΓA have trace zero, except the identity Γ0.

Trace Norm

Hodge* dual map
maps entities to pseudoentities and vice versa

CLm(n)  →  CLn-k(n)

This operation is Qset theoretical cognate of taking the relative
complement

A  →  S - A

Duality Rotations: exp( i α e(n) )

Clifford conjugation
maps a vector x to -x

Automorphisms of Algebras

Involutive Automorphisms
Order 2
Order 4  Fourier

Automorphisms of Clifford Algebras

Inner Automorphisms

Outer Automorphisms

The Inner product on the Clifford algebra

```
Spinors Cf. Spinor Contexts
```
Spinors are found algebraically within a Clifford algebra as 1-sided
ideals, so generally, there are two classes of spinors of given
dimension.

The simply connected spinor groups that are the univeral covering
groups of the doubly connected orthogonal and complex orthogonal groups.
O(n, C), SO(n, C).  These groups arise by analytic continuation of
the real parameters of O(n) and SO(n).  This is immediately possible
since Lie groups are always analytic manifolds, and the coordinate
parameters of these groups are analytic functions on the Lie groups.

The role of the bivectors in defining the planes of rotation, in V.

Relation of the spinor representation to the vector representation.

Finkelstein's Question: Does Cliffordization of a Clifford algebra
as a vector space yield a Clifford algebra?

Conceptually, the operation of Cliffordization can be viewed as a
"Fermionic Quantization"; e.g., Cliffordization of a 1-particle QM
projective Hilbert space is exactly a Fermionic 2nd quantization.
A solitary, isolated particle "does not know" whether it is a Fermion
or a Boson, i.e., which statistics it will obey.
The statistics, is as statistics is, a concept relative to,
and within a collection or population.

```
Closure of the bivectors on SO(n) ring
```
For an underlying Euclidean space, define the bivector basis

b[kj]  :=  [Γk, Γj]

which will obey the commutation relations

[b[kj], b[lm]]  = δkl b[jm] + δjm b[kl]

- δjl b[km] - δkm b[jl]

This provides a representation of the orthogonal algebra so(n), but
the matrices will be 2n x 2n.
This is actually a representation of the associated "spin group",
if the algebra is real.

For a pseudoeuclidean space simply replace the δ with the metric tensor,
for the representation of Spin(p, q), with p+q=n.

If the algebra is complex, the representation is that of su(n) and
is its spin representation.

```
Closure on bivectors with vectors on ISO(n) ring
```
As before, define the bivector set antisymmetric in its indicies
Inclusion of the vectors in the SO(n) ring forms ISO(n)
with the vectors Γi as generators of translations.

[b[kj], Γi]  =  δik Γj -  δij Γk  ?? check indicies

The Lie group ISO(n) is a semidirect product of the group of
rotations with a commutative group of translations.

-----------------
Systems defined and constructed by direct product
C & Q
-----------------

```
Construction from the C*-algebra A(V) of operators on V:
```
Now let us look from a different perspective, where we take the elements,
or points of S and quantize them individually, by associating to each
element a copy of FCCR(2) algebra of 2x2 complex matrices.

The Q rule for combining systems is to take a direct product; doing that,
we have an n-fold direct product of 2x2 matrices yielding a matrix algebra
of 2ⁿ x 2ⁿ matrices, where the algebra has dimension 22n.

The space V may be mapped by injections into the space of the
tensor product of CL(V) with itself:

V  →  V x V  →  A(V)  →  CL(V) x CL(V)

into the algebra as projection operators, in the tensor product of V
with its isomorphic dual space, and so this last construction is
actually that of CL(V) x CL(V), a direct product of CL(V) with its
isomorphic dual as a vector space.  This direct product realizes
the C*-algebra of linear operators on CL(V) as a vector space.

-----------------

This shows that quantization of the power set of S, mapping the Boolean
ring to a Clifford algebra is equivalent to quantizing the singletons
of A individually, and forming the direct product as Q composition of
systems, provided that the singleton quantization is a map of each point
to a copy of the algebra M(2) of 2x2 complex matrices.

Either from the periodicity theorem or from the above,
however, it is seen that this map to M(2) for each point
must be the mapping of V to V x V to A(V) to CL(V) x CL(V) where V is of
dimension 1. and so the M(2) is actually a single "point" that must always
carry along with it a peculiar charge that accounts for its own possible
state.  This evokes the idea of an algebraic form of a
membership function concerning a point and a potential singleton set.
[I believe David Finkelstein said a number of things very much like
this over twenty years ago, and I'm sure that he talked about the
bracing operation.  He also considered the construction of sets and
their cardinalities through

{} → {{}}, {{}, {}}, {{}, {}, {}}, ...,

thus replicating the vacuum, and taking about an algebra with the
operations of bracing and replication, thus, in a way, creating
something from nothing in grand Wheelerian fashion.]

The passage from V to CL(V) mirrors a bracing operation which takes
an element x to its singleton {x}; and in a Q context the distinction
between x and {x} is physically enforced as a real physical concept,
since the possibilities of both {x} and {} must be taken into account;
the null set is associated with the space of scalars.
This is perfectly in keeping with the Q principle seen in Feynman's
path integral formulation where all *kinematical* possibilities are
counted and weighted, whether or not they are classically, dynamically
possible.

Moreover, a point acquires even more structure since it now comes
equipped with some dichotomic variable that behaves much like the
spin of the electron, or like the electric charge of isotopic spin;
the mathematical formalisms of these two are identical.  So, points
acquire a strange kind of "charge" when quantized.  The difference
here, however, is that the spin algebra is complexified, so that
instead of a three dimensional su(2) algebra, we have an eight real
dimensional M(2) algebra which is a gl(2, C) algebra to describe a
Qpoint with irreducible quantum extension resulting from the
incompressibility of the Planck extensions of space and time.

GL(2, c)  =  D X SL(2, C), D is a complex dilitation.

```
The Physicality of It All
```
Again, the important aspect of any concept of "quantum set theory" is
that it is fundamentally a model of a *physical* concept, and not just

Physical structure becomes more explicit, e.g., when FCCR(2) expressing the
Canonical Anticommutation Relations (CAR) is written in terms of the
Fermionic creation and annihilation operators F† and F, respectively,
in the "number representation", where

│ 0   1 │
F  =  │       │,  and F† is the Hermitean conjugate
│ 0   0 │

so, the anticommutator

{F, F†}  =  I(2), the 2x2 identity.

If {x-} is associated with (1, 0)† and {x+} with (0, 1), then
(topologically, think perhaps in terms of oriented 0-simplex)

│ 0   1 │ │ 0 │     │ 1 │    │ 0   1 │ │ 1 │     │ 0 │
│       │ │   │  =  │   │    │       │ │   │  =  │   │
│ 0   0 │ │ 1 │     │ 0 │,   │ 0   0 │ │ 0 │     │ 0 │

F : {x+}  →  {x-}  annihilating a unit charge from x
{x-}  →  {x0}  annihilating x by annihilating its charge
while,

│ 0   0 │ │ 0 │     │ 0 │    │ 0   0 │ │ 1 │     │ 0 │
│       │ │   │  =  │   │    │       │ │   │  =  │   │
│ 1   0 │ │ 1 │     │ 0 │,   │ 1   0 │ │ 0 │     │ 1 │

F†: {x+}   →  {x0} annihilating x+ by attempting to create an
additional unit charge.  [Formally, think of a
binary field where 1+1=0.  Galois fields have an
implicit toroidal global topology.]
{x-}   →  {x+}  creating a unit charge for {x-}

NB: the null set {} = {x0} is a vacuum, but it is still a "thing",
a kind of container for a quantum point with its acquired charge.

Construct the Q(2) and P(2)
-----------------
```
Reconceptualizing Set Theory
```
Originally, S contains points and one can freely select those
points to form subsets, not worrying about point duplications,
and assuming that one can always distinguish otherwise instinguishble
points by some labeling procedure.

Now, in forming subsets, one has to be careful about selecting
points that fit into a priori measured subset containers whose size must
also be specified, size being the number of holes that can be
occupied by points.  If a container's size is k, the number of
points m that can be placed in the container must be m ≤ k.

Basic Combinatorial Problems:
If the holes are distinguishable
If the points are distinguishable
If the points are indistinguishable
If the holes are indistinguishable
If the points are distinguishable
If the points are indistinguishable
-----------------

Raise the states into the FCCR algebra and generalize to density
matrices.

The quantum set theoretical states are actually represented elements
of the algebra, and so the algebra of operators "the old observables"
are actually elements of a
CL(n) X CL(n)  =  CL(2n).
For one Qpoint, that is CL(4), this is the complexified Dirac algebra.

-----------------

This means that an irreducible "quantum point" has a structure and that
the structure is formally expressed by M(2), which is spanned by the
generators of the Lie algebra gl(2, C), of the complex general linear
group GL(2, C), of real dimension 8.
The (Fermionic) "quantum point" is then a quantization of a classical,
orientable 0-simplex.

-----------------
For a diploid set, the appropriate algebra is CL(4), which is a
complexified Dirac algebra.

-----------------

Physically, then, an abstract quantum point of any set is a Fermionic
entity to which the Pauli exclusion principle can be applied: the fully
antisymmetrized "states" (products of the anticommuting Γk)
corresponding to the classical subsets of S, but are also typical of how the
states of Fermionic systems combine in quantum theory.

The Pauli exclusion principle can be applied to such quantum points
once the symbol of energy can be defined.  Pauli's principle says
that Fermionic objects of a *system* can only exist in distinct
energy levels of the system.  This is the ruling principle for the
counting procedure in quantum statistical mechanics that results
in Fermi-Dirac statistics.  To use this principle here requires the
construction of Fermi-Dirac statistical mechanics of quantum points.
In order to do this however, it is necessary to specify the form
of single Qpoint energy, and possible interaction energies so that
the energy levels of the system and their occupation numbers can be
computed.

Without even getting that far, by looking at things very generally,
Pauli's principle has an interesting consequence: by so severely
restricting the occupation numbers of system energy levels of systems
(quantum sets) of Qpoints, the principle implies that that system will
not, generally, be able to undergo a condensation phase transition.
This means that Qpoints maintain their own proper extensions, as
well as the extension of a set or ensemble of Qpoints.

An important point to keep in mind, structurally, is that
Bosonic condensation is an ideal phenomenon in the limit
of an infinite number of points/particles.  Aspects of
any recogizable phase transition are only seen within the
context of a "sufficient" number of whatever statistical entities.
Size may not matter, but number does - that, in itself is a
strange and very interesting idea that connects what we do
in our pathetically limited intelligence with physical reality.

One can still, however, think of the structure of "Cooper Pairs"
in the theory of superfluidity that allows Fermions to pair off
as antialigned entities of net zero spin that become Bosons that are
capable of such a condensation, allowing the physical development
of what one might call singularities.  A little speculation allows that
if one wanted to see physics geometrized so that particles arise
from spatial structure that this kind of pairing and condensation
may be a useful idea by explaining that particles are essentially
condensates.  But, interestingly, in both theory and practice,
to see a Bose-Einstein condensation, a fairly large number of
particles is required to exist in the sample under inspection.

If there is a fundamental size associated with a Qpoint, say the
Planck length, not only will it be impossible for Qpoints to coalesce,
there will be a lower bound on the size of Qsets of n Qpoints for
any n.  This insures that Qsets always have an extent that can be
interpreted as physically spatial, as well a temporal, as we shall
see.  To say the least, it matters whether physical points of space
are Fermionic, Bosonic, or otherwise.  That, simply put, singularities,
i.e., infinities in the spacetime fabric appear not to exist,
regardless of the pathologies possible in physical theory predicated
on continua, is a wrong argument in favor of a Fermionic quantum
nature of physical points.  This means that a reasonable first
approach to the subject at hand in physical terms is through
Clifford algebras.

First, let s be a general element of CL(n), representing a quantum state
of an abstract set of classical cardinality n.

The general "pure" quantum state s of the set S is a complex linear
combination of the 2ⁿ antisymmetric basis elements above, and as
such can be parsed into a double sum,

n (n m)
s  =   Σ   Σ  ck(m) Γk(m)
m=0 k=0

where k labels the basis elements spanning the subspace with m
products of the n defining generators Γk, and m labels those homogeneous
subspaces.  The cm are the complex coefficients, and Γk(m) are the
basis elements.  In the uniform algebra basis defined above

2ⁿ
s  =   Σ  cA ΓA
A=0

Then
Tr( ΓA s )  =  z(n) cA 2m, for n = 2m, 2m + 1
where
z(n)  =   2m, for n = 2m, 2m + 1,

and where A labels the uniform basis elements spanning the algebra CL(n).
Furthermore,

2ⁿ
z-1(n) Tr( s s† )  =   Σ  cA cA†
A=0

2ⁿ
=   Σ  |cA|²
A=0

The "pure states" (a density matrix M, where M² = M) of a set
is then an abritrary collection of subsets with
complex weighting, that could be normalized to provide a probability
interpretation in the usual way, and in so doing, passing from a complex
Clifford algebra CL(n) to a projective complex Clifford algebra PCL(n).
The subsets of the fixed set S are its natural "eigenstates" specified
by the basis ΓA.

A collection of subsets of S is a generator of a topology in S.

What does a subset mean physically?  It means that the elements
of the subset have formed a system, and are then in some sense
in physical proximity to one another.  The state s expresses the idea that
the elements of a quantum S can partake of membership in several
subsystems of the system S, in interfering quantum alternatives.

A probability interpretation for CL(n) demands that

n (n m)
Σ   Σ  ck(m) ck(m)*  =  1
m=0 k=0

where '*' indicates complex conjugation.
Then, properly we must as done with the Hilbert space in QM,
pass to a Projective Clifford algebra PCL(n), whose elements are
the rays of CL(n).  The above normalization condition selects
representatives of the rays as classes of states, and these
representatives are those lying on the Euclidean unit sphere of
the 2ⁿ dimensional vector space that is CL(n).

The reversion Automorphism and the dual algebra of linear functionals

```
The Number (Cardinality) Operator
```
On CL(n), and then also on PCL(n) as (projective) vector spaces, define a
linear operator # on the g-basis that maps every element Γk(m) to (m Γk(m))

# Γk(m)  =  m Γk(m)

Defined on the basis elements, its action extends to CL(n) and to
PCL(n) by linearity.

```
Expected Cardinality
```
n (n m)
Σ   Σ m ck(m) ck(m)*  =  <s|#|s>
m=0 k=0

```
Representation Constructions.
```
See [Hestenes 1966], and Brauer & Weyl [Brauer 1935]

Quantum Cognates of Classical Set Theoretic Operations.

There are two levels of quantizations involved

First quantization maps a set Sn of cardinality n to a complex
Hilbert space Vn of dimension n.  The partition of Sn given by
singletons {x} of all the elements x in Sn is mapped to an
orthogonal basis of Vn.  Such a mapping is then invariant
under the group of complex transformations of basis that
preserves orthogonality since no particular basis is preferred.
That group is is the Lie group U(n).  U(n), in addition to
preserving angles, also preserves lengths of the vectors of Vn,
meaning that for the purposes of first quantization, the lengths
of vectors of Vn may be factored out of the situation by
standardizing them, and associating with each of orthogonal
basis elements vj an equivalence class {cj vj} where cj
are arbitrary non zero complex numbers, taking as a representative
subclass those elements with norm 1.  This still leaves a phase
factor free so that the equivalence class of vectors {exp(iaj) vj}
is associated with a point xj of Sn.

Ordering points of Sn and equivalence classes of basis vectors of
Vn:

For any finite n, the elements of Sn can be labeled with a set
of any n consecutive integers, as can the elements of a basis of
Vn, hence their equivalence classes.  Given the integer set, this
can be done in n! ways, and from these possibilities any particular
permutation of labeling that might be convenient can be chosen.
There is nothing in any conception of quantum theory to prevent
this.  Regardless of which set of integers is chosen, or how the
integer labels are permuted, this choice determines an ordering
of the xj.

Projectivizing, so that

Sn  →  PVn
by
xj  → { exp(i aj) vj }

where j ranges over the set of consecutive labeling integers.

This still being "first quantization", there is only one point,
and the Sn actually represents a set of possibilities that
can be associated with that point; this is a classical cognate
of the quantized situation.  It is important to have such cognates
in any quantum theory because an essential part of quantum theory
has always been a method by which quantum things my be interepreted
in classical ways in order that there be predictable classical
results that can be subjected to measurement.  It is important
to recognize that any act of physical measurement is the result
of some subtle aspect of interaction between a quantum system and
a macroscopic (non quantum or classical) measuring device, by which
the object is forced briefly to exhibit a classical aspect.

No other recognition in standard quantum theory shows so forcefully
how quantum theory is a construction whose very purpose is to
describe one aspect of reality in the language and terms of another,
under the condition that the aspect to be described defies the
terms and language by which it is described.  From this condition
comes the unending seeming craziness and confusion about quantum
theory.

The points of S are classical possibilities of a single Qpoint, and
are more like binary holes that may be filled or not filled by a
"dynamical Qpoint".  What is characteristic of any quantum theory
is the quantum states are complex linear combinations of the
classical possibilities (outcomes of any classical determination
by measurement).  Here, already, the essence of a constraint set
by a model of physical reality enters the picture.

Introducing the Dirac notation for State Spaces

A "quantum state" of a Qpoint constrained to be an element of the
set Sn as a "ket vector", in Dirac notation, as,

|f>  =   Σ ak |vk>
k

Any Hilbert space, and also its associated projective Hilbert space
is an inner product space isometrically isomorphic to its dual
space of linear functionals; any vector |v> of PVn then has a
dual, denoted by a "bra vector" <v|, so that the "bra(c)ket",
elision produces the notation,

bra ket  :=  bra(c)ket

<v| |u>  :=  <v|u>

which denotes the inner product of |v> and |u>.  The map from the
space kets to the dual space of bras in any representation,

|v>  →  <v|

is accomplished by Hermitean conjugation, which is to say the
application of the commuting operations on matricies of complex
conjugation and transposition.

Since |f> is a member of the projective Hilbert space PVn,

<f|f>  =   Σ (ak)* aj <vk|vj>  =
k,j

Σ (ak)* aj δkj  =
k,j

Σ (ak)* ak  =    Σ |ak|²  =  1
k                  k

having used the orthonormality of |vk>.

This allows the |ak|² to be interpreted as probabilities.  If
the Qpoint is "in the state |f>", then a classical measurement
of "where the Qpoint is", must have as an outcome on the
possibilities of Sn.  The probability that the Qpoint will be
found in the hole xk of Sn is given by

|<f|vk>|²  =  |ak|²

The complex quantity <f|vk> is called a "probability amplitude".
Such an interpretation can be easily extended by asserting that

|<f|g>|²

is the relative probability that if the Qpoint is in the state
|f>, upon a determination of the state, the state will be found
to be |g>.  This assumes, of course, that determining the state
of the Qpoint can be done; but this assumption of possibility is
already implicit in the saying of "let the Qpoint be in the state
|f>".  Methods of such a preparation and of subsequent
determination are not required to be the same.

in many texts either not, mentioned at all, are thereafter ignored,
are not made any physical sense of, or worse, obfuscated.  In
discussing the interpretation of |<f|vk>|² and |<f|g>|², the
notion of succession enters in such a way that the bra is understood
to preceed the ket temporally.  It could, of course, be the other
way around since

|<f|g>|²  =  |<g|f>|²

The associated amplitudes, however, are complex conjugates of one
another.  While the norm, <f|f> of any element of PVn is real,
and has the value 1, the inner product <f|g>, an amplitude, can,
in general, only be understood as complex, so

<f|g>  =  <g|f>*

In real spaces, one can write for an inner product

x · u  =  |x| |u| cos θ
or

cos θ  =  x · u / (|x| |u|)

for the cosine of the angle between vectors x and u.
In a complex space, let

|f>  =  |x> + i |y>
|g>  =  |u> + i |v>

where |x>, |y>, |u>, |v> are all real vectors, then

<f|f>  =  (<x| - i <y|) (|x> + i |y>)
=  (<x| - i <y|) (|x> + i |y>)
=  <x|x> + <y|y> + i (<x|y> - <y|x>)
=  <x|x> + <y|y>
=  1

<g|g>  =  (<u| - i <v|) (|u> + i |v>)
=  <u|u> + <v|v>
=  1

<f|g>  =  (<x| - i <y|) (|u> + i |v>)
=  <x|u> + i (<x|v> - <y|u>) + <y|v>
=  <x|u> + <y|v> + i (<x|v> - <y|u>)

:=  r (cos θ + i sin θ)

where

|<f|g>|²  =

r²  =  (<x|u> + <y|v>)² + (<x|v> - <y|u>)²

With

|<f|g>|²  =  |<f|f>|² |<g|g>|² cos² α

cos α  =  |<f|g>|
=  r

for the angle between |f> and |g>.

The phase of the amplitude is given by,

cos θ  =  (<x|u> + <y|v>)/r

1
=  -----------------------------------------------
(1 + (<x|v> - <y|u>)²/(<x|u> + <y|v>)²)1/2

or

sec² θ  = (1 + (<x|v> - <y|u>)²/(<x|u> + <y|v>)²)

But

sec² θ + tan² θ  =  1

so,

tan θ  =  (<x|v> - <y|u>) / (<x|u> + <y|v>)

We also have then the relation,

cos α cos θ  =  (<x|u> + <y|v>)

```
Intermezzo On Classical & Quantum Statistical Mechanics

Ontology, Representations & General sets

Ontology and Set Theory

Consider a universal aggregate U of monadal (in the sense of Leibnitz) entities, all identical replicates of one another. If we can somehow isolate one of these entities in some way, we have committed the first and primary act or operation of human cognition: we have made a distinction or discrimination. This is this, and that is not this: the foundation of classical logic. Both such distinctions, and inability to make them is intimately intertwined with the thought processes of quantum statistical mechanics.

This should lead us to examine the concept of ontology, and perhaps understand it not as a monolithic concept, but one that needs to be understood in terms of a hierarchy that transcends the well accepted Aristotelian metaphysics abstracted and generalized from formal logic.

When we insist on mixing physics with metaphysics, something quite unavoidable in the pursuit of science, avoiding such levels and hierarchy of ontology is equally unavoidable, and yet the mixed language of such things tends to avoid some obvious necessary distinctions of the hierarchy.

Formally, we include the possibility of this primary cognitive act in our models of reality by assuming for the models the Law Of the Excluded Middle (LOME). A HS teacher long ago typified its usage with the discriminating assertion, "everything in the world is either a flea, or not a flea." It is a primary axiom of the model of standard symbolic logic, but this is not the same as the Law Of Discrimination (LOD) that allows us to make the discrimination in the first place. Notice that a discrimination has just been made, not only between two different laws, but also implicitly between two types of laws.

The LOEM is merely a law for symbolic manipulation, while LOD which necessarily comes before it, asserts a structural relationship between some cognitive model of reality and human cognitive capabilities that operate on that model. The uneasy assumption is that this cognitive construction somehow mirrors some aspect of the reality presumed to lie beneath the sensory perceptions upon which the cognitive model is built.

The threat of circularity in this primary concept of discrimination is very real. We prevent (or complicate) the issue of circularity by making such distinctions as just made, and these distinctions are really distinctions between levels of ontology.

The most available example of these ontological distinctions is the general distinction between theory and metatheory, prototypically between mathematics and metamathematics, properly understood between physics and metaphysics.

It is often necessary to split naïve concepts in order to avoid either circularities or paradoxes. An early example of such a split is the Richardian paradox arising from the naïve concept of "the set of all sets", can be argued to be a subset of itself, and not a subset of itself, and therefore paradoxical. The problem, of course, is the language which foolishly confuses the concepts.

A solution is to bar the expression defining the concept as stated, and insist that it be properly framed as "the aggregate of all sets", or "the category of all sets", the point being that the concept for which "aggregate", ot "category" is a label, is not the same as a set. A label is the same conceptually as a coordinate value. The word "category" has, of course, has been selected as standard in the literature for the concept. Similarly then, one may not speak of the "categories of all categories" either, but is forced to split off another concept, and so necessarily speak perhaps of "aggregates of categories".

In principle, this process creates a denumerable infinity of hierarchies of language, requiring the linguistic form "(n+1)-gories of n-gories", but never "n-gories of n-gories". This is to say that, in a roundabout formal way, that the concept of existence is not a unical concept, but necessarily, a concept of hierarchies.

To say that "X exists" is meaningless, unless the statement is also placed within a hierarchy of levels of existence. Language "exists", but it does not exist on the same level as elementary particles are said to exist. Their "existence" is ambiguous, at least and even within the usual concepts of quantum field theory. The existence of language, presumably on a higher level is even more suspect, and so we become skeptical of, and more particular in language in describing any level of existence. We must not equate a thing with a symbol of a thing, much less a theory of a thing.

This is metalinguistics, where we speak of the structures of formal linguistics, but we want to return to the levels of the total picture that lie beneath the levels of formal languages. Those two levels are a putative core ontology, and the ontology of the cognitive abstractions that we generally assume mirror the core ontology in some way, however badly.

Again, we make a distinction within the usual ontological garbage can: there are levels of ontology. A thing "Ding an sich" may be said to have core ontology; my cognitive abstraction of it may also be said to have an ontological property, but these two ontologies are entirely different, and practically speaking, these two different ontologies would be almost impossible to correlate meaningfully, and must reasonably and logically be held separate.

The ontology hierarchy gets kicked up another notch should we begin to notice that human communication is matters such as these can to some extent be shared. Is there then an ontological level of which each of our individual cognitive experiences is merely a projection? This is after all the level of the Platonic ideal.

It may be that this is a worthwhile level of ontology to create or conceive of. In doing that we have immediately operated within the level meta to it, and in my writing about, I have gone to the metameta level. Do my metameta considerations have ontology? Clearly they do in terms of quantum neurochemistry, and the proof is the mirroring of that you are reading, as these are glyphs encoded by electrical bits, nybbles and bytes.

As the concept of language necessarily splits into the formal hierarchy of language levels, the concept of ontology splits necessarily into a parallel hierarchy of "n-tologies".

But, is this not all a matter of formal nonsense wherein the n-tologies do not "really" exist? No, exactly because each of them as conceptually constructed has a physical presence in terms of quantum neurochemistry, and in that sense they have a physical existence and manifestation. The warning, however, is still perfectly clear that n-tology is not equivalent to m-tology, when m ≠ n. The only way to avoid this hierarchy is to deny 0-tology, and so deny all of existence. That would be a major impasse, and less than useful, reducing all language and "knowledge" to random babbling, whose existence (0-tology) could not be allowed. Maybe that is exactly what it is, but we seem to have indications and feedback to the contrary.

Is the hierarchy of n-tologies really infinite as it would appear? No, the ultimate cap being the finiteness of the universe, though the cap of a significantly more finite human provides an earlier cutoff.

Infinitudes or finitudes, this begins to look almost like the perverse worlds of German idealism as expounded upon by Hegel, where all "real" existence becomes pure thought. It is, of course, a flavor neoplatonism. Platonism can be a useful and aesthetically satisfying frame of mind, unless one begins to take it seriously, which, alas Hegel did.

The major difference, as far as I can see, is that Hegel thought of existence as something absolutely monolithic and without a hierarchy. Here, however, a hierarchy of ontologies is exposed as being a necessary aspect of the concept.

One can make similar hierarchical distinctions among: thought, language, and the written symbols of language if only because their ontologies are also quite different - and even more complicatedly interrelated.

So, what has all of this to do with set theory?

Underneath the formalistics of set theory lies a hardwired human cognitive apparatus, of which the formalistics are an abstraction. We make distinctions, a primitive and also powerful operation that can lead to considerations of ontology as above. That was really all about making necessary distinctions.

The presumed ability to make the single isolation of a "one" from the others, even if only in our heads, gives the foundation of the conceptualizations of both set theory and the natural numbers as naïve concepts. It provides the foundation for the construction assessment of measures, of patterns, pattern matchings or recognitions, differences of patterns, and finally time by utilizing memory in combination of sequences of pattern differences. The perception of time is a cognitive activity; we create our mathematical models of time based in this primitive and subconscious construction.

To a large extent, the assumption of the ontology of time, as we perceive it, is a consequence of our neurochemistry of perception; we have great difficulty in conceiving of other precisely because we are awash in this construction like the proverbial fish in water.

We are also awash in distinctions even more primitively, and primitively we immediately tend to interpret the world around us in terms of the cognitively processed sensations available to us. There is in this the same understandings and some of the insights of Kant in his "Critique" as well as that of several modern neurologists, among then Thomas R. Damasio. While Protagoras may well have expounded well with philosophical insight that man is the measure of all things, it is more truth than philosophical poetry, for whether or not he is aware of it, man's given self is that by which he does measure all, all that he is able to measure at any given time. The ability is not an absolute.

It is an evolutionary trick that works well in the context of the mesoscale world of his existence, but which is bound to error when his sensibilities are extended to worlds much smaller, and to worlds much larger. The difficulties of the scientist in entering conceptually into these other worlds are surely that they defy the immediate sensory perceptions with which they come equipped and indeed programmed; a fair amount of irrational imagination is required of any theoretician.

Playing on the distinctions first, I believe, made by Plato, Oscar Wilde said, "Man is a rational animal who always loses his temper when called upon to act in accordance with the dictates of reason."

What was in back of Wilde's rather prodigeous mind in saying that, other than as usual to induce thought is beyond me. But, by saying it, he creates a dichotomy of the rational versus the emotional, a distinction which is both enlightening and obscurant.

The obscurant aspect is to suggest that emotions are somehow extralogical, and that they are chaotic, which, of course, is entirely untrue. The differences between what we call reason and emotion depend rather on the differences in premises rather that the procedures, progressions and sylogisms.

If one can understnd that the logic of emotions exists, one also then understands the logic behind the behavior or animals other than human, and so understand that "being a rational animal" in no way distinguishes homo sapiens sapiens.

What truly distinguishes man from other animals is not that he is rational, but his ability to be willfully irrational and willfully quite insane. Man is not the "the rational animal"; if any thing he is "the crazy animal", that routinely acts completely contrary to his own best interest, often from ignorance and misguided application of idealism rooted in Platonic thinking.

Other animals might be said to be "crazy" only insofar as their relationsships with "man" are considered. On their own, they do behave according to their natures and best interests. A dog's, or cat's life is not exactly ideologically motivated, which is not to say that abstraction and prediction is beyond them. They are mammals, and the reptilian brain is already capable, and even dependent on such pocesses.

While this characteristic insanity is clearly responsible for every myriad misery, horror and atrocity of human history, it is also responsible for all his arts amd sciences. All artists and scientists are just a little bit, or a lot, crazy; that goes with the field, as they regularly conceive of perfectly "crazy" things that "normal" people do not. That is exactly what they both do, and that is also exactly why they are valued.

The overarching problem is that Wilde's distinction is commonly held, and that n-tological distinctions are not. They are, however, intuitively made by artists and scientists. Wilde made equivalences precisely to indicate the falsities of language; he was not simply being cute or outré.

Before leaving Wilde and Plato, it might be worth noting that the context of n-tologies, the question of whether the idealized Platonic forms of mathematics, lines, points, triangles, tetrahera, etc., are real is fairly easily settled in the affirmative my noting that they are mental abstractions that necessarily have neurochemical existence. Now, play with associated equivalence classes, if you like, to make things tighter. In any case, we have just objectified ideal Platonic forms of whatever kind. N-tologies are really all about objectifications.

Back to Set Theory:
In classical set theory, we assume, even in U, the aggregate of replicates that somehow, if only conceptually, we may still label these utterly identical elements of U, and so create distinctions where none exist, except on a different level of ontology. If the elements have 0-ontology, their 1-ontology is being assumed to be equivalent to their 1-ontology. But this equivalence is not valid, and our quantum theories tell us this precisely, and this is the difficulty of understanding quantum theory: it defies the structure of our neurological hardwiring that gives rise to the 1-ontology of cognition.

Since Gödel, we have understood the importance of levels of langauage, and that metamathematics must be distinguished from mathematics, but, it seems that the levels of language regarding containers has not yet suggested the importance of the levels of language of that which is contained by the containers.

```
The general dictum derived has to do with distinctions:
Do not confuse your models with reality;, i.e., do not
fail to make that distinction.  This is already a dictum
in a meta meta theoretical language.

If we can isolate one element, there should be no especial reason
why we can do it again, which is to say, allow replication of the operation
of discrimination as the replication of the elements of U was allowed.

Question: Is the operation of replication universally, i.e., within
and among all levels of language, allowable?
Given a finiteness of the universe, is there a "within
reason" codicil?  How would that matter, formally?

Question: On what level of language should the previous question(s)
be considered?
```

If two isolations can be performed, and the two isolated monads can be labeled, say A and B, in containers labeled 1 and 2, there are two situations we can consider: (A1, B2) and (A2, B1), or reducing the notational language in an obvious way, (A, B) and (B, A).

A quantal way of thinking about the primitive isolation problem in U is to consider a one monad container, and ask whether or not it is occupied. Classically the answer is either yes (1), or no (0). Quantally, we assign a complex number z to the container, with the square of the absolute value |z|² ≤ 1, with z in the closed unit disk of the complex plane.

There is a seemingly redundant way of expressing this idea with two complex numbers z0 and z1, the first giving the amplitude for nonoccupation and the second giving the amplitude for occupation so that

```
|z0|² + |z1|²  =  1

```

This is the correct statement of the quantum model, and the mathematical model using only one variable was a cheat that structurally is no different from an unnecessarily complicated expression of a fuzzy set relation: there is no provision for the necessary phased competition in superposition of classically mutually excluding alternatives, which the LOEM does express.

Making a leap that will be obvious to any physicist, a quantum theoretic SU(2) structure is a model of a dichotomic variable is the replacement for the functional, classical LOEM. Physicists tend to think of the SU(2) structure and its attached "kinematical" CAR constraint as being the mathematics of spin, when it is really the correct quantum expression of any classically dichotomic concept, including the concepts of the LOEM and of the cognate set theoretical concept of "occupation".

The mathematics of SU(2) and CAR in quantum theory is an appropriate model for the basis of a quantum set theory of singletons in any U. The question to be posed and resolved within this primitive theory is then whether or not the container is occupied, or not. (Think perhaps of U as a thermodynamic reservoir to which the container in question is attached.)

One way of thinking about the transition from the classical concept to the quantum concept (so "metaquantizing" conceptually) is to understand that the question of occupation splits into two distinct, but related questions:

```
1 Is the container occupied?
2 Is the container empty?

```

A tacit assumption of this little discussion is that the isolating container can contain only one monad, and we can claim that this is a feature of this system, and not a bug.

The feature has an important consequence of allowing that the model can easily control/track cardinality of a set that is to be constructed from quantum singletons modeled by SU(2).

One might ask the question, do these sets actually exist? From the discussion above on n-tologies, the question should seem to be neither nonsense nor trivial. From that very same discussion, it should be clear that the answer is yes, but that one must be careful about distinguishing n-tologies. The care involved should not really distinguish between the classical situation and quantum situation; the appropriate questions need to be asked in both cases.

With all of the above in mind, consider an n-dimensional complex Hilbert space associated to (represented by) one monad and n-containers, with  > 1.

The very formal expressions of quantum theory express relationships between a quantum ontology (0-ontology) and our apparently classical, perception bound existence (1-ontology). If we were somehow quantum beings capable of cognition, our theoretical physics would then presumably look very different. An especial difficulty, and yet also perhaps our saving grace regarding epistemology and physical theory is that while in full structure our beings are classical regarding perceptions (i.e., we perceive classical classical structures), the underlying perceptual neurology has indeed a quantum level existence, as being chemical, it must.

One clue to the underlying Q nature of our neurology may be any of the now classic Gestaltist experiments where an image can be perceived in two different ways. Is it two profiles, or a vase? Anybody who has played with these things knows very well that seeing (perceiving) the two possibilities is an all or nothing experience. You see one, or the other, and with effort switch between them, after you have discovered them. You can never see the two possibilities "superposed", and in this sense, the model of observing a quantum dichotomic variable comes to mind.

It may be worth while to look at these two viewpoints in more detail. The viewpoint that yields a single complex number isolates the single container and ignores its conceptual complement altogether. The alternative of what happens to a monad if it does not occupy the container is not explicitly accounted for. The viewpoint that yields the two complex numbers explicitly considers this alternative.

A rather general principle of quantum theory, often reinforced by Feynman, is that a quantum theoretical model must enummerate *all* (kinematically) possible states of the system. The first viewpoint does not do this; the second viewpoint does, and that is why the second is the correct formulation.

Warning: the Hilbert space construction following immediately is a cheat in exactly the same way that the single complex number model was a cheat.

The complex Hilbert space Vn can be constructed as an n-fold tensor product of the occupation amplitude z ∈ C. With the classical structure of mutual exclusivity of the containers for one monad, we naturally impose on the independent amplitudes, in parallel to the normalization condition on the SU(2) situation of mutually exclusive classical alternatives,

```
n
Σ |zj|²  =  1
j

```
where j labels the independent copies of C in the tensor product. If the occupation of each container is a statistical matter, then so is the overall matter of occupation in any of them. That the theory imposes this normalization constraint is an indication that it very specifically looks to the classical nature of the classical side of the relationship expressed by quantum theory between quantum and classical ontologies. The standard normalization is then not simply a technical convenience of probabilistic formalism; it is an expression of the fact that there is a classical aspect of interpretation contained in the quantum theory. That is to say that intrinsically, quantum theory is not just about quantum ontology, but is really about a quantum ontology that is related to what amounts to free ranging classical measurements: it is about a set of quantum kinematics of possible states, a quantum dynamics of sequences of states, and a connection between quantum states and classical measurements on those states. A reading of von Neumann [Neumann 1932] makes this clear from another viewpoint, simply by distinguishing the unitary transformation of temporal propagation of the state function from the nonunitary propagation of measurement.

This points out a very difficult and confusing aspect of quantum theory: it necessarily contains a connection between the "bizarre" quantum world and the structure of the human sensory world of classical physics.

The formal method of combining systems, valid in classical and quantum contexts is by direct (tensor) product.

For one monad with n containers, we will need one copy of SU(n). This nicely generalizes the SU(2) model above since one of these containers can always be taken to be the container that is "someplace else". If we do that, then there are really n-1 containers. Let k = 0, 1, 2, ..., (n-1) label the containers (or states); we always let k=0 label the "someplace else state". In a very real sense, this is what Dirac's original "hole theory" was about.

One can also interpret the SU(n) model as a single quantum monad with n-ary (rather than binary SU(2)) logic.

For n monads with one container that can contain at least n monads, SU(n) is appropriate, but for accounting for a monad not being in the container, and that requires some messiness: a direct sum of SU(p) x SU(q), with p+q=n. There are (n 2), a binomial coefficient number of elements in the direct sum.

For two monads each in n containers, the cheat is resolved by taking a direct (tensor) product of SU(n) x SU(n) instead of SU(n) because there are independent yes-no classical choices for the occupation of each container.

Hilbert space of two monads and n-containers Same containers v. different containers.

The cheat is resolved by taking a direct (tensor) product of n copies of SU(2) instead of C because there are independent yes-no classical (dichotomic) independent choices for the occupation of each container. For one monad in n containers, therefore, we will need a tensor product of n copies of SU(2).

While the monads are quantally indistinguishable, the question is whether or not the containers are indistinguishable.

For m monads in n containers an m-fold product of SU(n)

Alternative interpretations? For one monad in n containers, we will need a tensor product of n copies of SU(2).

SU(n) with n being the number of containers or the number of truth values.

direct (tensor) product as a

If points of a quantum set theory are quantum objects, it appears that we must make a decision as to whether they are Bosons of Fermions, ignoring parastatistics for now.

If one is considering the physical kind of space with which we are familiar, apposed to some abstract mathematical space, to begin with, one can argue in favor of the stability of space by assuming a fundamental Fermionic nature, since a Bosonic nature would presumably imply that low energy space would routinely suffer Boson condensations.

Even with a Fermionic assumption, however, remembering the BCS formalism of superconductivity and its "Cooper Pairing", a kind of condensation behavior is not completely locked out. If one thinks finitistically, such condensations will only look like the peaks of singularities, but not actually be singularities. Overall, then, making the Fermionic assumption for physical space seems to mediate undersireable extremes and would seem to provide the necessary physical behavior.

There is a difference of thought pattern here, however, that is almost the opposite from thinking in the BCS context. Spatial singularlities (or pseudosingularities) occur in regions of high energy density and not low energy density. So, one must have the idea that this condition is what forces the Fermionic enities together in such a way that they exhibit Bosonic properties. In GR, if one allows an energetic content to the spacetime 'substance' itself, essentially allowing a nonvanishing cosmological constant, then that idea is that a gravitational force of contraction overcomes the peculiar Fermionic force of repulsion, and so achieves the effect. Only the mathematics will know for sure, and seeing how the pairng mechanism works in BCS formalism would probably be a good test and clue to whether or not that story pans out.

```
Symmetry and Antisymmetry - indistinguishability.
Container exchange
```
```

Any set theory of elements is indifferent to what the elements
or points actually represent, so the vectors of Vn are also
indifferent to what they may represent.

Formal Set theory may be mapped to a formal propositional calculus;
A formal quantum set theory can be mapped to the formal propositional
calculus of quantum logic as discussed long ago by Birkhoff, Jordan,
von Neumann, et al., discussed in terms also of orthomodular
lattices, opposed to the Boolean lattices of classical logic.

Varadarajan and the Geometry of Quantum Logic

The Membership Relation

System Combination and the Direct Product.

Where do the permutational symmetries even (Bosonic) and odd
(Fermionic) come from in the context of indistinguishability of
points?

Second Quantization and Tensor Algebras

Symmetric and Antisymmetric Tensor Subalgebras

Supersymmetry and Fermionic Creation and Annihilation Operators

s in SO(n)

Inner         &  Outer Automorphisms

s Γk s(-1)  =  Σ Mkj Γj
j

s in SU(n)

```
Classical & Quantum Correspondences
```
Fermionic second quantization maps 2S, the power set of S to the
Clifford algebra CL(V).  A Table of conceptual correspondences
in this case:

Classical S                   Quantum, Hilbert V and Clifford CL
----------------------------------------------------------------

point                         a basis vector of V (eigenpoint)

singleton                     basis vector of a CL (eigenset)

set w/ cardinality k          subspace of V of dimension k
CL element of homogeneous degree k

(illusory, not ontological
but conceptualized intermediate
point)
general singleton             an element of V
CL element of homogeneous degree 1,
i.e., an l.c. of the generators.

irreducible partition         an orthogonal basis of V

complementation               orthogonal complementation in V
Hodge-star map in CL

Algebra deformation?          ------

│ cos a  -sin a │ │ A   │     Duality rotation
│               │ │     │
│ sin a   cos a │ │ Ac │

disjointness                  orthogonality
union of disjoint sets        tensor product
union
intersection of sets          intersection of subspaces in V
division

-------------------------------------------------------------------
Boolean Ring (associative)    Clifford Ring (nonassociative)

symmetric difference          Clifford anticommutator
Clifford IRREP associative
multiplication

∩ (intersection)

∪ (union)                     Tensor product

difference                    Associative multiplication

Lie multiplication (antisymmetric)

CL(n) as a Finite Fermionic Fock Space

Statistical Mechanics and Density Matricies
direct product CL(n) X CL(n) → CL(2n)

Cluster Expansion

The Dimension of subsets

The dimension of a k-point set must be less than k.

The Dimension of S

Expected Dimension

Automorphisms of a kinematical algebra and the most general
"dynamical evolution"

I will now resort to FCCR theory to develop an energetics of Qpoints,
and Qsubsets.

Let the Γk correspond to the eigenvectors of Q(n)
Projection operators
Density matrices.

Simplicial homology connection and the boundary operator

```
Hypergroups

Quantum Categories

If, in a general mathematical sense, one can quantize set theory, then a second quantization is a quantization of categories. That done, it seems, in a formal sense, that an entire body of quantized mathematics can be developed. Perhaps then, more generally, again formally, that an entire body of "deformed mathematics" can also be constructed, much as one speaks of deformed algebras since formal logic may be expressed as an algebra from a metamathematical perspective.

Nothing seems immune to the quantization monster. Yet, structural change by defomation appears to be easier and better defined that the physically illusive notion of quantization.

```

TOC

Footnotes

1.

2.

3.

Physics Pages

Math Pages

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Email me, Bill Hammel at

bhammel@graham.main.nc.us

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Created: August 14, 2003

Last Updated: June 26, 2004

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