Please Read to understand links and footnotes.
The two metaphysical principles that govern modern physics are the quantum principle and the relativistic principle, for each of which there are several mathematical formulations.
In various attempts to yoke these two principles together in terms of formulations one comes ultimately to the conclusion that there is a lower limit to which discrimination of points of space and time can be taken; these ultimate limits are usually expressed by the Planck units of length and time. [i] Thus, while concommittant formulations of both principles may separately be selfconsistent, with the included straightforward model of a continuum for space and time, their combination and simultaneous satisfaction is already conceptually inconsistent with that straightforward assumption of a continuum model.
This signifies that while the two principles themselves may very well be simultaneously and consistently valid in some overall model, their current mathematical formulations cannot be so implemented.
Moreover, the notions of internal symmetries are not only not covered by the quantum principle as currently formulated in terms of the Canonical Commutation Relations (CCR), they are also not generally conformable to the relativistic principle.
Additionally, while the quantum principle is expressed in terms of CCR and the Canonical Anticommutation Relations (CAR), these two expressions are not easily related to each other, except through the connected idea of permutational symmetries that are symmetric and antisymmetric respectively, in accordadance with Spin-Statistics theorem of quantum field theory. [w] A useful formulation of the quantum principle that unifies CCR and CAR is wanting. Though they look deceptively similar in algebraic statement:
CCR: q p - p q = i,
CAR: q p + p q = i,
their representation theories are vastly different. Intermediate to them, however, though not expecially helpful, is the theory of deformed algebras, and in particular, "quantum groups", which are not groups after all. Go figure. At any rate, consider the equation
q p + α p q = i, where α is a parameter with elements in a field that connects the two values α = +1 and α = -1. This generalized expression has both CCR and CAR as special cases, and α is called a parameter of deformation.
That's it; I'm not interested in doing an extended thing on quantum groups or supergroups here. The only thing to be said is that supersymmetry [w] is a way combining the two different kinds "symmetry" and "antisymmetry" associated with CCR (Bosons) and CAR (Fermions) respectively, which is very different from the combining relationships found here. Supergroups provide a context for combining algebraically CCR and CAR, while quantum groups (which are not groups) interpolate algebrabically between CCR and CAR, which is again, is very different from what is found here.
What follows is an attempt to explore the usefulness of the local and quantumly finite and discrete mathematical model FCCR(n) that can replace the kinematic foundation of quantum theory based on CCR.
TOC
If, below the Planck regime, points should not exist, as all current theory indicates, at the Planck level [i] one can expect the minimum discrimination, which would mean a "simple" two-point discrimination to be possible.
An important lesson that one learns from Quantum Mechanics (QM) [w1] [w2] regarding the formal representation of spatial points is that in any quantum theory that treats position as an observable represented by a linear operator, spatial points are collectively represented by the spectrum of an Hermitean position operator and the collection of the eigenvectors [w] associated to points of the spectrum. In QM this spectrum is routinely a continuum, and unless it is artificially, and I dare say unphysically and impossibly constrained by infinite potential barriers (and therefore infinite energies) to be otherwise, it is also unbounded. Although it may appear that something appropriate regarding quantization of space has been done in QM because in its abstract formulation the position coordinates are represented by operators, as is often the case, appearances can be deceiving: space and time in QM are exactly the Euclidean entities that they are in Newtonian mechanics. [w]
A twist in the case of QM as constrained by CCR is that eigenvectors of the position operator do not actually belong to the projective Hilbert space [w] of states; nevertheless, they can be used as a basis to represent vectors that are legitimate states by linear combination.
Aside:
While a separable Hilbert space always has a countable basis,
the eigenbasis, δ(x-x') of standard QM position operator is
uncountable; the relevant concept explaining this seeming
contradiction is "overcomplete basis", a concept that comes
into play more decidedly in the context of "coherent states".
Visually and notationally perhaps think of a diagonal, continuous matrix with the real line laid down the primary diagonal. Each point of this real line (an eigenvalue) is then associated with an eigenvector that is orthogonal to all the other eigenvectors. This, in one and any specific dimension, is a description of the QM position operator in diagonal form.
Using this position operator formulation and the idea of restricting observation to smaller and smaller intervals down to the Planck regime, one can ask what the mathematical formulation would look like. The easy and almost unavoidable answer is that "position" with this minimal two-point discrimination would most generally have to be a linear operator representable by a 2x2 complex matrix. [w] You need noncommutativity in an algebra of operators to express anything quantum theoretically and an algebra represented by 2x2 matrices is the smallest such thing possible.
Exactly such an expression is given by CAR, which has only one finite dimensional irreducible representation in terms of 2x2 matricies. Moreover, quantum phenomena represented with exactly this minimal algebra already exist multiply, e.g., electron, proton and neutron spin, isotopic spin of electromagnetic charges, and the statistical antisymmetry of Fermionic particles expressed as CAR.
Formally then, a simple formalism restricted to a regime of the Planck scale represented by 2x2 matrices reflects a structure of reality of which any concommittance of quantum and relativistic principles must partake, and interestingly, this is exactly the arena of any irreducible, finite dimensional representations of the CAR which, formally, thinking in terms of the Pauli exclusion principle [w], holds such a pair of points, whatever they may be physically, apart.
It is probably no accident that this matric algebra M(2, C) [w] supports the defining representations of the Lie groups [w] [i] GL(2, C), SL(2, C), SU(2), SU(1, 1), SO(1, 2) and their Lie algebras, [w1] [w2] and that SL(2, C) is the universal covering [w] group of the orthochronous, special Lorentz group, and its isotropy subgroups. [w1] [w2]
If there is indeed a key to fundamental physical theory, it is in M(2, C). The statement may be new in form; the particular argument may even be new, but the idea in almost ancient. The attraction is that it is the simplest possible context where principles of Quantum Theory (QT) and Relativity Theory (RT) [w1] [w2] can and do operate concommittantly. This is also the only currently known context of such concommittance.
Proceding from this to anything interesting is the difficult part.
"The difficult we do immediately;
the impossible takes a little longer."
-- Alleged Motto of the US Army Corps of Engineers,
also attributed to the US Navy seabees,
also attributed to the US Air Force,
also attributed to the Mauritius Police Force;
it's apparently very popular among those who can
do neither. I guess I'm safe.
If one understands that it is inappropriate to use a naïve continuum model for physical extensions like space and time, the idea of formulating any fundamental physics in terms of differential equations is out of the picture; algebra, which expresses relationships is in the picture and becomes the focal point.
With no particular physical principles available below the Planck regime regarding measurability, except that their existence doesn't seem to make much sense, the only mathematical clue is that the fundamental language should be algebraic and that this language should somehow give rise in a natural way to Lie algebras. Within a general context of algebra there is a considerable amount of mathematical space to explore.
The antiquity (fifty years ago :-) of the idea that the continuum is in physics a problematic aspect of its models once quantum QT had made its relevance felt is suggested by Einstein's writings, e.g.,:
"Adhering to the continuum originates with me not in a prejudice,
but arises out of the fact that I have been unable to think up
anything organic to take its place. How is one to conserve
four-dimensionality in essence (or in near approximation) and
[at the same time] surrender the continuum?"
-- Albert Einstein
Einsteins's Reply to Criticisms in Relation to
Epistemological Problems in Atomic Physics
from The Library of Living Philosophers Series (1949)
How much "discreteness" appears in the model may, to some extent, be a matter of computational convenience that takes into account that at least part of any Q theory seems to require and accomodate a classical continuum of real probabilities:
If one assumes the "possible in principle" when really, it is not possible, is shown by both Q and R principles that this can be a grave mistake. Both principles are restrictions on what can be known, and what can be done. Q theory, in particular, tells us that the Planck regime should be quite real, and yet, the obvious restrictions contained in this concept have no fundamental place in the existing mathematical formulations of Q theory; it should.
It is clear that no physical measurement can ever be made with the precision demanded to distinguish arbitrarily close real numbers. For this understanding, classical physics is sufficient, but not necessary. That is obvious and we use real numbers anyhow; they are convenient simply in matters of calculation. Even if we used in theory only rational numbers, within that confine, perfect discrimination is still a fantastic ideal, yet these, or rather a subset of them, are the numbers written in any finite number of laboratories.
Understanding an intrinsic quantum discretation of reality does not mean that all continua in the mathematical model must be abandoned, but it does mean that the fundamental limitations must be built into the model so that we can easily distinguish and respect those limitations and tolerances from the continua of computational convenience.
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The complete 2x2 matrix algebra over the complex field [w] also happens to represent, as its only irreducible representation (IRREP), the Clifford algebra [w] of subspaces of an E3, with its standard inner product, in a sense extracting from the simplest form of a general quantum theory appropriate to the Planck regime, a local three dimensionality of space. This indicates that the essentially three dimensional nature of space, at least with a relatively low energy density, has a quantum theoretical origin.
A small aside on dimension:
We say that we live in a three dimensional space, and somehow that makes
an intuitive sense even to people who have no idea what a mathematician
or physicist might mean rather more precisely by that. Physicists, in
fact, have a rather precise meaning that is rather crude from a
mathematician's viewpoint. To a physicist who thinks in terms of phase
space or configuration space, dimension is simplistically synonymous
with the number of independent degrees of freedom;
there are problems with this idea, as the existence of Peano
curves and Hilbert cubes formed bt space filling curves indicate.
Mathematicians, on the other hand, have many different approaches to the concept, or what might be called the problem of dimension [Hurewicz 1948]. The most common concept of dimension is the inductive definition of what turns out to be the topologically invariant dimension first outlined in 1912 by Poincaré. In 1913, Brouwer brought Poincaré's idea to formal rigor. That was forgotten until 1922 when Menger and Urysohn each separately and independently of the previous work rediscovered the same inductive idea which logically refines the intuitive idea of physicists. The idea is based on the observation that intuitively you need a 2-dim space to divide a 3-dim space into two nonintersecting parts; you need a 1-dimensional space to do the same with a 2-Dim space, and you need a 0-dimensional space (a point) to do the same for a 1-dimensional space. Generally the, an n dimensional space is a boundary of an (n+1)-dimensional space.
.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
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.
But there are also inner measures, outer measures, the Carathéodory and Hausdorff measures that lead to concepts of dimension based in measure theory rather than topology. From Hausdorff dimensions then comes fractal dimension. [Mandelbrot 1977]. Such dimensions, unlike the Brouwer topological dimension are not necessarily integral.
In the various theoretical approaches and meandering about the great
unification problem stated in the first paragraph here, it seems that
the "real" or at least appropriate topological dimension of spacetime
(if you believe
in that) can reasonably be assumed to be 10, 11 or 26 dimensions in the
directly topological sense of Brouwer. Yet the dimension of the universe
as a space over which matter in distributed as a Hausdorff measure
is rather less than 3, by seemingly empirical studies.
One cannot take a simple or seemingly obvious answer to the question of
the dimension of physical space because the definitions are varied
and because the these different definitions are convenient in
different contexts (i.e., partial or incomplete viewpoints) of
physical reality.
End aside.
If any form of a quantum theory is necessary to assert at least a three (Brouwer) dimensional existence, I would claim for many reasons (not gotten into here) that without the Q essence of physical ontology, existence itself (as we know it) would be impossible, and that one must somehow have this macroscopic three dimensionality of space, automatically. A passing remark: topologically, it is really four dimensions that turns out to be the richest of all, and also the dimension for which various results and proofs of topology fail, presumably because of that.
It is not unreasonable, however, to conjecture that a high energy density may increase a local spatial dimension. The simple thought pattern is that stuffing things into something requires it to expand. That this is perhaps unexpectedly limiting in terms of dimension may be seen by computing the volume of an n-ball of fixed radius r, and seeing what happens to the volume over a significant range of dimension. For convenience, I give the appropriate formula for the volume of an n-sphere of radius r:
Vn(r) = √πn rn / Γ( n/2 + 1 )
The standard Gamma function behaves like a factorial [w], which dominates the power function of (√π r)n, and so that in the limit of large n, the Vn(r) approaches zero. For a fixed r, Vn(r) rises to a peak in the volume for some n, then Vn(r) falls off to zero as n→∞, by the dominant factorial growth of the Γ-function.
Perhaps not quite what might be expected.
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Ninteenth and twentieth century physics takes place in a three dimensional manifold [w], or in a four dimensional manifold as the given mathematical models of space and and spacetime, so it would be good to say a few words about manifolds. Loosely speaking, a manifold is a topological space [w] (a set of points together with a certain class of sets called a topology) that locally looks like some Rn. Additionally, there are continuous, bijective functions [w1] (homeomorphisms) [w2] that tie together and relate the local spaces.
It is easy to consider such a "flat" En being either honeycombed or covered with a collection of sets. An easy honeycombing for E² is a triangulation or decomposition by simplexes [i]. that will exist for any En, as will also, any decomposition by n-orthotopes, the generalizations of squares and cubes. [i]
A great difference between these classical manifolds and what might be called "quantum manifolds" is that, for a classical manifold, while a patch is always the same patch, and the manifold exists independently of its covering or honeycombing, these statements are not clearly valid in the context of more generalized structures where the structure of patches themselves as well as their relationships can change. An accumulation of the past in any such dynamical structure would reasonably be expected to involve a quantum entanglement of patches. [w]
If a patch corresponds to an algebra, a subpatch corresponds to a subalgebra, so a set theoretic intersection of patches corresponds to a common subalgebra, while a set theoretic union of disjoint patches cooresponds to a direct sum of algebras. The more general union of two patches with nonnull interesection needs the idea of interlocking of algebras developed below.
To each patch algebra there is also a representation (REP) of the algebra, i.e., an algebra homomorphism mapping the algebra into an algebra of linear operators realized as complete matric algebra, which is in some sense a "patch state", within which there are also "patch processes" most generally represented by density matrices. Think, e.g., of angular momentum or spin whose values prescribe IRREPS of so(3) or su(2) repectively. Though, for any su(n), there is always an infinite number of finite dimensional REPS, in physical truth they cannot all be available.
There are REPS that are irreducible representations (IRREPS) and those that are not. For any given algebra, the IRREPS span the ring of all REPS. and thus the space of all states of a generic patch is a "linear combination" of IRREPS, stressing their importance as fundamental entities. By weighting the IRREPS by elements of a field, the ring is transformed into an algebra of REPS.
Processes in/of a patch correspond to formal quantum states, i.e., to density matrices that form a cone in the representing matric algebra, the boundary of which is a space of pure processes, which space is isomorphic to the projective Hilbert space that is the carrier space of the representation. These processes are quantum processes relative to a state of a spatial patch, thus bringing together the quantum nature of the container and the contained.
System in Patch (The container and the contained)
Cf. quantum set theory II
A kinematical algebra KA(m), that takes the place of CCR kinematics
(possibly FCCR(n) - which generates the (n) IRREP of su(n) by commutation)
The expression of "Dynamics" means specifying energy operator E(m),
as algebraic expression of the elements of KA(m), which E(m) specifies
T(m) by a Fourier transform. This unites with the KA(n) to form the
dynamical algebra DA(m).
NB: the phase time of T(n) is *not* a uniform Newtonian, nor a full
blown relativistic proper time coordinate. It is purely a local to
patch quantity. T(n) represents an extension or degree of freedom
of any quantum system, and one which is Fourier related to a local
concept of energy, by a specific Ansatz in the local FCCR model of
quantum theory. [i]
One can think of this phase conceptually as one can think of the gauge
invariance of the first kind in QM: "invariance under local phase
transformations". The locality, i.e., local independence of these
transformations mean that they are anholonomic. In elementary gauge
theory, the anholonomicity is "explained" by associating the
indeterminate phase θ with a line integral of the electromagnetic
potential.
x
θ(x, x0) = - e ∫ Aμ dxμ
x0
'e' in this case being the charge of the electron.
The phase indeterminacy is then explained by the presence of an
electromagnetic field F, derived from the potential by an exterior
derivative.
Fμν = ∂/∂xμ Aν - ∂/∂xν Aμ
When F vanishes identically, any loop integral (x=x0) also vanishes,
showing by Stokes Theorem that the phase function is well defined at
every point.
Now think of the Newtonian concept of time progressing *uniformly*
at each point of space; this uniformity is two fold, that somehow
time units do not change in the passage of time, and also that the
progression is the same for all points of space. Relativity theory
explicitly denies this uniformity of progression over spatial
extension, and that is a definite can of worms, in good measure
due to the failure to distinguish the various meanings that are
confused in the very overloaded, hence ill defined and conceived
word "time". See [Kaempffer 1965], sects. 20,21,22.
See also [Origins of the Species of Time].
Kaempffer calls gauge fields "compensating fields", and illustrates the
idea by showing the gravitational field of the Einstein Equations, and
Yang-Mills Fields to be compensating fields that are nonlinear and
nonabelian in contrast to the electromagetic field, remarking on their
vector and massless characteristics.
In thinking of any fundamental physical theory as essentially local:
any derived relativistic time coordinate and relativistic relations *must*
be given in terms either of expectation values or of transition amplitudes.
Just a flash insight. Suppose for example the ST vector is given as a
Cartan matrix, which is really a displacement from the origin.
Elements of a local Hilbert space H(n) specify pure system process.
H(n) x H(n) in M(n, C) specify general density matrix processes,
DM(m, C).
Space System in Space
M(m, C) ←───── DM(m, C) ←───── H(m)
↑ │
│ ↓
patch ──────→ algebra ──────→ REP ←─────── DA(m) ←─────── KA(m)
(size n) su(2) e.g. ↑ │ │
│ │ │ │
│ │ │ │
└─────────→ IRREPS E,T ←── f ──── E
For su(n>2), not every integer > n has an IRREP of su(n).
E.g., there is no (4), (5), (7) or (9) for su(3). While su(2) should
be sufficient if space is all there is, the inclusions of particles
and their "internal symmetry spaces" means that higher unitary
symmetry algebras must be included and allowed, if a generalized
geometrization program in the spirit of Riemann, Clifford and
Einstein is to be pursued. [3] See below.
XYZZY Space as a homogeneous space? ISO(3, 1)/SO(3, 1) = E⁽⁺⁺⁺⁻⁾, i.e., Minkowski space Local spatial cyclities and their breaking: cyclic C(n) v. shift ш(n)
IRREPS as pure patch states (processes) from which general patch states REPS are then constructed by weighted direct sums. a general patch state is a weighted direct sum of REPS, then the Clebsch-Gordan coefficients and their generalizations to a greater number of REP direct multiplicands, are special in this context, and in the rings or algebras of representations of an appropriate class of Lie algebras, [w] e.g., the rings of representations of all su(n). These coefficients give the weights in a reduction of any REP (usually given as a direct product of IRREPS) to a weighted direct sum of IRREPS.
So far, in very general terms, there is a structure corresponding to a topology to a quantum manifold connected to a lattice [w] (I have not shown this explicitly in the present context for good reason) of subalgebras of a largest algebra representing the manifold as largest patch, together with a state of the quantum manifold given by the collection of states of all the patches represented by the collection of REPS attached to the nodes of the lattice.
The lattice picture (locally) comes from thinking of space as a covering by patches which discretize any local model as an Rn that would correspond to the local picture of any manifold.
The open sets of a topology form a lattice. A lattice of submodules of a module partially ordered by inclusion is a modular lattice. Through lattice theory, a possibly desireable connection to be made is with pointless topology and topoi. [w] Some other time maybe.
There is enough freedom in the choices of REPS attached to nodal patches that a putative local quantum manifold structure as given by local lattice structure can be broken by the REPS. E.g., if P1 and P2 are two patches of the lattice with lattice partial ordering
P1 ≤ P2
meaning that P1 is a subpatch of P2, so for the corresponding algebras A1 and A2, A1 is a subalgebra of A2. One would expect then that S1, the state of P1 would be a "substate" of S2, the state of P2 in the sense that there exists a projection operator acting on the REP R2 determining S2 that can project out the subREP R1. Suppose that by isomorphisms,
A1 = su(2)
A2 = su(3)
and that R2 is the defining (3) IRREP of su(3), while R1 is (4) IRREP of su(2). Then, the required projection operator cannot exist, since one cannot project in this way a 4 dimensional space *into* a 3 dimensional space; so, the simple and consistent subpatch relationship that ostensibly appears in the model may be, to some extent, a matter of computational convenience which takes into account that at least part of any Q theory must accomodate a classical continuum of probabilities:
In the next section, we can look at another algebraic way of talking about an En in terms of its subspaces of dimension k ≤ n, by looking at the precision demanded to distinguish arbitrarily close real numbers. Specifically, we look at doing this with an E3 using a Clifford algebra. [w]
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The dimensions of the subspace types, names of the associated tensorial forms, and spanning algebra elements of the this Clifford algebra are: [w]
dim 1 scalar (magnitude) σ0 dim 3 vector (length) σ1 σ2 σ3 dim 3 pseudovector | bivector (area) σ2 σ3 σ1 σ3 σ1 σ2 dim 1 pseudoscalar (3-volume) σ1 σ2 σ3
letting σk, k,j=1,2,3, which span the vector subspace, be the generators of CL(3).
NB The element of any Clifford algebra labeled with a subscript 0 always turns out to be represented by an identity matrix of appropriate dimension.
The complete 2x2 matrix algebra over the complex field also happens to the algebra gl(2 , C), the defining relations of the generators are written:
{σj, σk} := σj σk + σk σj = 2 δkj σ0
where the δ is the usual Kronecker delta equal to zero unless k=j, space. This indicates that the essentially three dimensional nature in which case δkk = 1. The vector basis elements, σk, are of space, at least with a relatively low energy density, has a isomorphic to the Pauli spin matrices, and they also satisfy the commutation relations of the Lie algebra so(3), as well the Lie algebra su(2) of its universal covering group SU(2).
[σj, σk] := σj σk - σk σj = i εjkl σl
where the epsilon, ε is the usual completely antisymmetric tensor equal to zero for any equality of indicies, +1 for any cyclic permutation of (jkl), and -1 for any single exchange of indicies (anticyclic permutation).
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A vector in the Pauli-Clifford algebra is always represented by an Hermitean matrix of trace zero. Choose any one of these, and label it the position operator Q(2). Any Hermitean 2x2 matrix M will have two distinct real eigenvalues a and b, and two mutually orthogonal eigenvectors. The general annihilating polynomial of M is,
(M - a)(M - b) = 0
or
M² - (a+b) M + ab = 0
showing that the trace and determinnt functionals of M are,
Tr( M ) = a+b, and Det( M ) = ab
so that
M² - Tr( M ) M + Det( M ) = 0
as a special case of Vietå's theorem specialized to quadratic polynomials, relating the roots of a polynomial to it coefficients.
Dropping the dimensional indicator, a Q necessarily satisfies the quadratic with a Q of trace zero, i.e., a = -b, and then
(Q - a)(Q + a) = 0
or
Q² - a² = 0
showing that Q squares to a multiple of the identity.
The full matrix algbera M(2, C) of complex 2x2 matrices has real dimension 8. The linear subspace of Hermitean matrices has real dimension 4, and the linear subpace of Hermitean matrices of trace 0 has real dimension 3; there is then a one to one correspondence between an R3 and the space of Hermitean 2x2 matrices of trace zero. [2] One expression of this correspondence is
┌ ┐
│ +z (x - iy) │
(x, y, z) ↔ │ │
│ (x + iy) -z │
└ ┘
Compressing the notation by defining M(v), so
v ↔ M(v)
where M(v) is called the Cartan matrix associated with vector v. It is easy to show that the determinant,
Det( M(v) ) = - (x² + y² + z²) := - |v|²
so that
Q² = |v|² I
For the general eigenvalue problem for M(v), to determine eigenvalues m1 and m2, and eigenvectors |s1) and |s2), so that
M(v) |sk) = mk |sk)
For k=1,2, it becomes clear by standard methods, that the eigenvalues of M(v) are +|v| and -|v|, and the eigenvectors (unnormalized) are
┌ ┐ ┌ ┐
│ (x - iy)/(+|v| - z) │ │ (x - iy)/(-|v| - z) │
b₊ │ │ b₋ │ │
│ 1 │ │ 1 │
└ ┘ , └ ┘
where b₊ and b₋ are arbitrary. The eigenvalues are also easily
deduced from the values of the determinant and trace.
If the b coefficients are defined by
b₊ := √[z(+|v| - z)/(x - iy)]
b₋ := √[z(-|v| - z)/(x - iy)]
then the eigenvectors of M(v) become
┌ ┐
│ √[(x - iy)/(+|v| - z)] │
√z │ │ = |s1)
│ √[(x + iy)/(+|v| + z)] │
└ ┘
and
┌ ┐
│ √[(x - iy)/(-|v| - z)] │
√z │ │ = |s2)
│ √[(x + iy)/(-|v| + z)] │
└ ┘
with the two |sk) orthogonal with respect to the usual Hermitean
inner product.
(s1|s2) = z √[(x + iy)(x - iy)/((+|v| - z)(-|v| - z))]
+ z √[(x - iy)(x + iy)/((+|v| + z)(-|v| + z))]
= z √[(x² + y²)/(-|v|² + z²)]
+ z √[(x² + y²)/(-|v|² + z²)]
= z √[-(x² + y²)/(x² + y²)] + z √[-(x² + y²)/(x² + y²)]
= 2iz
(s1|s1) = z √[(x² + y²)/(+|v| - z)²] + z √[(x² + y²)/(+|v| + z)²)]
(s2|s2) = z √[(x² + y²)/(-|v| - z)²] + z √[(x² + y²)/(-|v| + z)²)]
One way of approaching and introducing spinors due to Elie Cartan [Cartan 1946] (there are several other ways), is to associate spinors with null vectors (also in this context sometimes called isotropic vectors), which have zero norm. Context must disinguish this "isotropy" of a vector from the "isotropy" of a space, meaning that there is no distinguished direction.
This is a bit of black magic, since in any Euclidean space, if the norm of a vector |v| vanishes, then v = 0, identically, there is nothing of any interest happening. In a pseudoeuclidean space like Minkowski space of special relativity, the matter is quite different, and lightcone vectors are perfectly respectable animals.
The black magic happens by inexplicably allowing that the R3 that represents the E3 is in fact embedded in a six real dimensional complex space C3, and that somehow it is assumed "physically ok" to rotate the E3 from its R3 position in C3 to some other linear subspace E3 - but - and here is the sleight of hand - ignoring the usual conventions for computing lengths of vectors in complex spaces that involve complex conjugation. In effect, the complex nature of the new coordinates is on one hand understood, but then also ignored, for no given or apparent reason other than to achieve the mathematical result.
The simplest example of the difference uses one complex number z = x + iy, where the modulus |z| is defined within
z z* = (x + iy)(x - iy) = |z|² = x² + y²
but
z z = z² = (x + iy)(x + iy) = x² - y² + i 2xy
which is something very different.
The magic show:
Perform the following rotations of R3 into C3
(x, y, z) -> (x+iy, x-iy, z) -> (x, iy, z)
This is an opposite of what would be called a Wick rotation [w] in field theory that maps a pseudoeuclidean Minkowski structure to a Euclidean field theory. The idea there is that the underlying assumption of analyticity (holomorphy actually) enables the easier results to be obtained in the Euclidean context and then analytically continued to the pseudoeuclidean context. The concept of analyticity arises naturally here in the context of Lie groups since a Lie group is simultaneously: a group, a topological space (that is also endowed with a left invariant "Haar measure" [w]), and an analytic manifold.
Now, compute the standard Euclidean norm, assuming that it is
analytically continuable, and set it to zero:
x² + y² + z² = 0
(x + iy)(x - iy) + z² = 0
x² + (iy)² + z² = 0
x² - y² + z² = 0
which describes a cone in a three dimensional pseudoeuclidean space, much like a lightcone of Minkowski space of Special Relativity (SR).
Aside: What the physics is behind this allowance that physical space
really does have such a complex extension of coordinates is a genuine mystery.
It certainly suggests that most generally Minkowski space should be
physically extended to C⁴, of 8 real dimensions, which can then be
parametrized by the full matric algebra M(2, C).
Then, of course, the questions of how one should back up to a seemingly
observable real subspace
from the complex space and why that should make sense
become important questions.
See [Classical Geometry & Physics Redux].
Pushing on to General Relativity (GR) then, the Einstein equations
naturally extend to a Kaehler-Riemannnian space presumably equipped
with a spinor structure.
End Aside.
With this constraint of zero norm, define,
u := √( -(x - iy) )
w := √( +(x + iy) )
then,
- (1/2)(u² - w²) = x
- (i/2)(u² + w²) = y
u w = z
A two component spinor associated with an isotropic vector v can then
be defined in column form as
┌ ┐
│ u │
|s(v)) = │ │
│ w │
└ ┘
and it is easy to see that, for an isotropic vector v,
M(v) |s(v)) = 0
From the square roots, which are double valued, a given isotropic vector has two possible associated spinors.
By standard considerations of linear algebra, for this special case of the eigenvalue equation to have a solution, i.e., for a |s) to exist that is not identically zero, it is necessary that Det( M(v) ) = 0. But that is exactly the isotropic condition. This last eigenvalue equation is the special isotropic case of the general one whose solution is given above, [i] where |v| = -|v|.
This introduces the definition of a pair of spinors
± |s(v))
associated with a E3 vector. It becomes clear, when the rotations of v mirrored by associated 2x2 complex matrices that concommittantly must rotate |s), that one cannot simply choose the signs for |s) as one often can in other contexts of taking square roots. This ambiguity of sign in spinors does not mean that the signs are intrinsically irrelevent, it merely means that relative to the E3 mapping to spinors that they are indistinguishible.
If you go back to thinking about the full complex matric algebra M(2, C) as representing the the Clifford [w] algebra for E3 as an inner product space, then, if "y" is a vector componet, "iy" is a pseudovector component, which makes very little sense. The M(2, C) algebra, however, can be seen as representing either The Clifford algebra CL(R3) of subspaces of a Euclidean R3, or the Clifford algebra CL(C²) of complex C² which is of 4 real dimensions. This equivalence is another way of viewing the homomorphic connection between between the Lie groups SO(3) and SU(2), and the isomorphism of their Lie algebras.
While the elements of the two Clifford algebras are the same, their interpretations differ:
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The tensorial explication of CL(R3) where the scalar field of the algebra is R, is above.
Yet, one can also notice that the general Hermitean 2x2 matrix can be associated with a 4-vector
┌ ┐
│ (t + z) (x - iy) │
(x, y, z, t) ↔ │ │
│ (x + iy) (t - z) │
└ ┘
again compressing notation for 4-vectors as we did for 3-vectors
x ↔ M4(x),
and
Det( M4(x) ) = t² - x² - y² - z²
which is left invariant along with its Hermiticity under the
transformations
M4(x) -> A M4(x) A†
with A in SL(2, C), the universal covering group of the proper
(Det = 1), orthochronous (t > 0) Lorentz group SO(1, 3). The
full M(2, C) algebra then decomposes accordingly as a real algebra
into Hermitean and skewhermitean subspaces, spanned by:
σ0 σ1 σ2 σ3
iσ0 iσ1 iσ2 iσ3
The 't' coefficient of σ0, however, is not a Clifford vector component
but a scalar, this is obviously not a faithful representation of the
Clifford algebra of a 4 dimensional real space, which has dimension
16, though there is a relationship to it. The relationship can be
seen in several ways, one of which is through the Brauer and Weyl
construction of Clifford algebra representations [Brauer 1935].
The Dirac algebra, on the other hand, is the appropriate Clifford
algebra for a 4 dimensional space.
For CL(C²), with complex scalar field:
dim 1 scalar σ0 dim 2 vector σ1 σ2 dim 1 pseudoscalar σ3
The reason the Clifford algebra interpretation has to be gotten clear is that spinors, one way or another, are associated with and connected to Clifford algebras since they are the elements of the carrier space upon which the algebra acts as an algebra of linear operators.
The four component spinors of the relativistic Dirac equation are acted upon by the elements of CL(C⁴) that is generated by 4 gamma matrices γμ, with μ = 1,2,3,4 with anticommutators satisfying
{γμ, γν} := γμ γν + γν γμ = 2 δμν γ0
NB: In a complex Clifford algebra, the signature of an underlying inner product space is lost, i.e., if we had started with the Lorentzian signature (+++-) or (---+), there would be no difference to the resulting Clifford algebra if we had started with an Euclidean E⁴. This is not true in the context of real Clifford algebras. [w] The representation theory of real Clifford algebras is more complicated than that of complex Clifford algebras. Fortunately, the Clifford algebras that appear in the context of any sort of quantum theory are necessarily complex because somehow any quantum theory requires the field of its algebraic expression to have complex structure in order to express those completely typical "quantum interferences of alternative states". Cf. [Mackey 1968], at 3.8, "Why The [quantum mechanical] Hilbert Space Is Complex".
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Tolerance spaces were, I believe, first introduced by E.C. Zeeman in 1961. Cf. [Zeeman 1962] The following closely follows that introduction.
A relation α between two sets X and Y is a subset of the Cartesian product X x Y. The inverse relation α⁻¹ ⊂ Y x X is the set of all (y, x), such that (x, y) is in α. If β ⊂ Y x Z, the composite relation αβ is the relation consisting of the the pairs (x, z) of X x Z for which there is a y in Y, such that (x, y) is in α and (y, z) is in β.
If A ⊂ X, denote by αA the subset of Y consisting of all elements related to elements of A:
αA = {y: y in Y and (a, y) in α, for some a in A}
A tolerance, ξ on X is a relation on X that is:
i. reflexive, (x, x) is in ξ
and
ii. symmetric, (x, x') in ξ implies (x', x) is in ξ.
The addition of the transitivity property,
(x, x') in ξ and (x', x'') in ξ implies (x, x'') is in ξ,
of course, would make it an equivalence relation.
A tolerance space, (X, ξ) is a set X together with a tolerance relation ξ defined on it.
For example, let X be a metric space with metric d(.,.), and let ξ be the pairs of points (x, x') with d(x, x') < ε, some fixed epsilon ε.
A tolerance on a space X induces a tolerance on the "usual lattice" of its subsets LX.
For A, A' ⊂ X, write the relation A =ξ= A' (A is ξ-indistinguishable from A') if A ⊂ ξA' and A' ⊂ ξA. Then the relation =ξ= is a tolerance on LX.
[What is really wanted here is an algebraic formulation of quantum tolerance spaces.] QPoint closeness is eigenvalue difference. for (+|v| - -|v|) = 2|v| = l0, the Planck length close to 0? We map a 3-vector Cpoint v to M(v) to a Qpoint pair each with eigenvector and the eigenvalues ±|v|. The map from M(v) to the eigen quantities is *not* linear. [i] This means that adding vectors in a resultant is not correctly mirrored in adding the |sk) associated to the vectors to get the |sk) of the resultant. The situation is not relieved for spinors attached to isotropic vectors.[i] For any two vectors v1 and v2 in R3, their inner and outer products can be expressed by anticommutator and commutator of their Cartan matricies respectively: (v1, v2) = (1/2) {M(v1), M(v2)} M(v1 ˆ v2) = (1/2) [M(v1), M(v2)] Note that M(v1 ˆ v2) is skew, and not Hermitean, and of course M(v1 ± v2) = M(v1) ± M(v2) then M²(v1 ± v2) = M²(v1) + M²(v2) ± {M(v1), M(v2)} = |v1|² + |v2|² ± 2 (v1, v2) As before, this last equation gives the eigenvalues of M(v1 - v2): ±d := ± √( |v1|² + |v2|² - 2 (v1, v2) ) = ± √( |v1|² + |v2|² - 2 |v1||v2| cos θ ) the angle θ being the angle between v1 and v2. If v1, v2 represent two arbitrary points in E3, (v1 - v2) is a vector that connects them, for which one can write M(v1 - v2) |d1) = +d |d1) M(v1 - v2) |d2) = -d |d2)
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If, metaphorically, we pull back a bit, allowing the region of discourse to widen, a third point can be distinguished from the quantum soup, and so the model will become a 3x3 Hermitean matrix.
A physical fact is that our "region of discourse" will always be finite, and so our representing operator will always be further represented by some finite dimensional nxn matrix, that exists in an algebra of nxn matricies. The content size of the region is measured by the dimension n of the algebra, while the state is measured by the dimension of the representation of that algebra.
However, once we have admitted three points, there are automatically three subalgebras to consider, which pair a point representation (a, b, c) into possible pairs (a, b), (b, c), (c, a), which must correspond to subalgebras of the algebra of 3x3 matrices. Think of the composing edges extracted from the boundary of a triangle in the context of simplicial homology.
Think also, e.g., of the three interlocked su(2) subalgebras of su(3). Representationally, they would have to be interlocked since each of the three point pairings shares a point with the other two pairings.
Enlarging the region of discourse further, there are four points, (a, b, c, d) which break up into triplets (a, b, c), (b, c, d), (a, c, d), (a, b, d), (faces of a tetrahedron) corresponding to, e.g., four interlocked su(3) subalgebras of su(4).
For su(4), each of the representing triplet subalgebras will have three interlocking su(2) subalgebras.
For su(n), there are (n n-1) = n interlocked su(n-1) subalgebras of su(n), where (n n-1) is a binomial coefficient. Each of su(n-1) has (n-1 n-2) = n-1 interlocked within su(n-2) algebras, and so forth, continuing the reduction down to su(2) subalgebras.
Equate n of su(n) with the number of verticies of a simplex. That means an (n-1)-simplex. For su(3), a triangle, with three su(2) line segments each joined to 2 others.For su(4), a tetrahedron, with 4 su(3) triangles, each joined to 3 others. The number of interlocked su(2) algebras is the number of line segments of the simplex, i.e., 6 for su(4), and generally, for su(n) then, there are n(n-1)/2 interlocking su(2) subalgebras, in which each one dimensional complex subspace of an su(2) carrier space is shared (n-1)-fold.
The multiplicity of sharing of subspaces, which is the uniform weight of any vertex of the geometric simplex as graph (a complete graph) is a dimension of a simplex. [i]
The n(n-1)/2 number of interlocking su(2) subalgebras of su(n) also happens to be equal to the dimension of the so(n) algebra.
As we pull back further and further, as some sort of idealized limiting deific observer, the size of the region of discourse is necessarily limited by the size of a finite universe that is measured by some very large n that is the dimension of an all encompassing algebra, that can be understood partially and vaguely as an algebra which represents a hierarchy (a lattice) of physically spatial subsets that contains the geometrical information of physical space; that is to say that the algebraic structure represents or models the topological information that implies the model geometry of physical space.
With an unbounded universe, pulling back in a limiting process will have a nilpotent Heisenberg algebra that is a fundamental kinematic assumption of QM, as the limit of the algebra. As per [FCCR III] the model limit should probably be understood in the strong operator topology.
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XYZZY 8-fold way, Gelman phenomenology. An organizing principle.
The symbiosis of symmetries and groups of transformations is well known in mathematics, but for the most part, the explication or expression of symmetries in physics is more in terms of Lie algebras than their associated groups. The reason for this is not exactly clear, but I would like to say a few words about this having to do with an intrinsic preference for not really needing continua in the statement of physical principles.
The fundamental equations of physics have, de facto, and traditionally since Newton, been given as differential equations. (Interestingly, they are all, in one way or another, statements of the conseervation of "energy".) It is more than interesting to examine what this locality means.
The relationships expressed by differential equations are local relationships, very local relationships; the solutions to these differential equations ellucidate the possible resultant global structures that are implied by the differential equations.
More specifically, the local relationships expressed by differential equations are of a "infinitesimal" nature. If one believes that infintesimals are somehow logically appropriate components of correct models of physically ontology (something I find to be empirical and experimental nonsense), then such expressions of fundamental physics though differential equations make perfect sense.
If one takes the alternative view that by their very necessary mathematical assumptions, differential expression of fundamental physical laws is already manifestly and empiricaally shown to be contrafactual, the only remaining mathematics left to express the appropriate fundamental relationships is algebra. The understanding though stumbled upon anew by me, was also given earlier by E. C. Zeeman. [Zeeman 1962] A truly fundamental physics must be expressed in an algebraic language that also expresses its fundamental Q randomness by certain Fourier transforms connecting eigenbases of selected operators in the algebra. These operators are most likely not to commute.
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Well before any concept of space and time upon which any Relativistic Principle is based, the very essence of reality is apparently well modeled, or at least restricted by a Quantum Principle that places limits on what can be known, indeed, limits on what even makes sense conceptually in any modeling of an a priori reality. Reality is frequently not amenable to our models, and this usually shows itself when our models contain assumptions of what is possible when it turns out that assumption is in fact invalid. More is possible than we thought.
Newtonian mechanics contains the implicit assumption that in principle a material particle can be accelerated to any unboundedly high velocity. From the viewpoint of the technology of the time, that wasn't even a questionable item; it turns out to have been false.
Both Newtonian physics and relativistic physics assumes that measurements or determinations of the values physical parameters specifying the state of a particle can be had with, in principle, an arbitrary degree of precision. Exactly, in principle, this is false.
Many attempts have been made to combine the formalisms of quantum theory with formalisms of relativity, not one of them being truly successful or even self consistent. Yet in every one there is the indication that below a Planck length, the very idea of coordinate or more simply, two point discrimination makes no sense.
Every major paradigm shift in theoretical physics has been one of restriction showing that what had previously been thought to be possible in principle was, in fact, impossible. More than that, the axioms of the new paradigm were statements that enforced the limitations that were newly recognized.
The problem is then how to state this essential limitation mathematically in the context of something like a dynamic manifold, which is exactly what this essay is about.
A truly penetrating formulation of Quantum Principle necessarily descends to the Planck regime correctly, something which the formulation in terms of the CCR does not do. Yet, the principle remains in the idea of uncertainty relations, usually stated in terms of a pair of purported observables. A question that arises is why should the fundamentals of existence be constrained formally by an uncertainty in two observables and not say three, where one might write something akin to
(UX) (UY) (UZ) >= const ?
For an explanation that is still obscure there is recourse to the idea of two point discrimination, or perhaps more to the point, the very foundations of concept of discrimination itself.
If one can accept the possibility that ultimately the standard concept of topological dimension remains valid in a better model of physical space, in any dimension n, assuming also the space's essential homogeneity (indifference to position) and isotropy (indifference to direction), the fundamental expression of discrimination between "points" is between exactly two points, no matter the value of n > 0. If any two points are held apart by some lower bound, this seems to guarantee that any n-simplex (of n+1 points) cannot degenerate to an object of dimension less than n, and so at least a contradiction of dimension is not given by a lower bound on two point discrimination.
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The special (Det=1) orthogonal Lie groups SO(n), n > 2, which express the rotational symmetries of Euclidean En, all have manifolds that are doubly connected, and there exists for each of these a singly connected covering group SPIN(n) whose set of irreducible representations (IRREPS) contains the IRREPS of SO(n), similarly for the respective Lie algebras. [Chevalley 1946].
Map of En to carrier space of the defining IRREP of SPIN(n) defines the space of spinors.
Continuing the IRREPS of SO(n) to SU(n) by complexification then also continues the IRREPS of SPIN(n) to USPIN(n) providing the spin representations of SU(n). [i].
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XYZZY The difficulties of generally relating topology and measure are not unlike those of relating homotopy and homology.
Coming soon, to a theater near you. :-)
Both topologies and measures are based on collections of sets closed under certain set theoretic operations, but the operations are different for topologies [i]. and measures. Topologies and measures can, however, be brought together rather naturally in the context of metric spaces.
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For a set S [w] of cardinality n, there is a power set of all subsets 2S, of cardinality 2n. The cardinality of all subsets of S of cardinality k has cardinality (n k), and of course, summing the cardinalities of the classes of subsets of cardinality k, there is the standard summation of binomial coefficients:
n
Σ (n k) = 2n
k=0
The set S of cardinality n can be quantized by a mapping from S to an orthonormal basis {|k>} of an n dimensional complex Hilbert space Hn. The algebra of subspaces of Hn can be made into a complex Clifford algebra [w] that is the quantization of the power set 2S by the mapping,
For k,j = 1, 2, ..., n
|k> -> Γk
where the Γk are the generators of the Clifford algebra CL(n), obeying
{Γk, Γj} = 2 δkj Γ0
Clifford algebras arise in any usual quantization of a collection of Fermionic objects, and so this particular quantization of set theory says that quantized subsets of S are Fermionic collections, antisymmetric under interchange of quantized points:
If k ≠ j, Γk Γj = - Γj Γk
The dimension of the Clifford algebra as a vector space is 2n, the same as the cardinality of the power set, its full basis being constructed by taking linearly independent m-fold products of the generators, m=0,...,n. For any such m, the dimension of the subspace spanned by m-fold products is (n m). The dimension of the algebra is the sum of the dimensions of these subspaces, which sums exactly the same as for the power set S, and the quantization of the family of subsets of cardinality m is the algebra subspace spanned by m-fold products of the generators.
Since the representation theory of finite dimensional Clifford algebras says that for any such formal algebra there is only one faithful representation in an associative algebra of matrices by anticommutators, the passage to unbounded n gives a Hilbert space spanned by a countably infinite (alef_null) basis, making the Hilbert space separable, and an associated Clifford algebra of dimension alef_one := 2^(alef_1) so that the Clifford algebra is nonseparable, with a dimension equal to the power of the continuum if the "continuum hypothesis" is invoked.
It may be worth remarking that while the basis vectors |k> represent points a in S, the Clifford generators Γk represent singleton sets {a} in 2S; that these are not the same things conceptually, is made obvious by considering a fuzzy set theory where element a belongs to a set only with some probability.
The space of 2-fold products, or bivectors of any CL(n) realizes the Lie algebra so(n) in its spin representation.
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Indistinguishability:
The first key to dealing statistically with large numbers of objects
in the context of quantum theory is that unlike the way it is done
in classical statistical mechanics (Maxwell-Boltzman), the objects
if the same, cannot be distinguished, even in principle, which is
to say that artifically distinguishing them with labels is forbidden.
Equivalent means strictly indistinguishable.
Symmetry and Antisymmetry:
The second key is a little stranger yet.
Odd dimension has scalar modes, even does not. Forces arise from virtual exchanges of scalar particles.
Permutation symmetry = Antisymmetric + Symmetric Gibbs - the other Gibbs
A superalgebra is usually defined as Z2-graded algebra. A Clifford algebra, by virtue of its even subalgebra, is a superalgebra.
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Since we don't really know what a point is, but do know about separations, say then that graphically, isolated points are not allowed. The smallest allowable structure is o----o, i.e., two vertices and an illusory connecting edge, which corresponds to the range of a smallest local position operator, su(2) in its lowest dimensional irreducible representation, i.e, its defining representation, which is the 1-dimensional simplex. After that,
o----o o
/ and / \
/ / \
o o-------o
are allowed, similarly allowed are all linear graphs:
o----o----o---- ... ----o
at one extreme, and all complete graphs (n-simplex) at the other, which
in three dimensions is a tetrahedron:
0
/\
/| \
/ | \
/ | \
/ | \
0/____|_____\0
\ | /
\ | /
\ | /
\|/
0
It is worth noting that the classical space of configurations of 4 bounded points is a 3-simplex, and similarly, that the classical configuration space of (n+1) bounded points is an n-simplex.
At least one sense of a local Newtonian microtime of a system 3 particles could be attached to measures of separation of the points of this 3-simplex. If one thinks relativistically, Einstein has already shown that the times of events cannot be ordered, except in relation to a given observer; so, trying to parametrize the events symbolized by the points of the simplex is a priori futile. A probability distribution over the points of the simplex can provide a kind of fixed and static dynamics for the system, on a classical level of statistical mechanics, but this is not a system with intrinsic quantum nature. On the other hand, specifying transition amplitudes connecting these points does have an intrinsically quantum nature.
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Point set topology anaylyzes a space by intersecting subsets, quantizing the situation analyzes a space by interlocking subalgebras. A "thing" never really exists in isolation, but only in the context of its environment, and the matter of isolating a subalgebra is a matter of physics and measurement, still just a model and not the reality. If a concept of spatiotemporal contiguity can be established, there are still the connections that transcend these connections which we usually call quantum entanglements that proceed from interlocks. If we believe in concentrated beginning of the universe, like the Big Bang, then it is easy to expect a great dominance of fundamental entanglements. A question, of course, is on which levels of ontology do entanglements exist? A plexus of such entanglements effectively increases the "dimension" of reality, and looks rather like Bohm's concept of Implicate Order. We don't understand "things" as the algebras, but rather as states or processes of the algebras, and connections, contiguities and entanglements by algebra homomorphisms. Is there, as it seems there may be, a fractal like structure to it all that gives form to a more general form of the old alchemical adage "as above, so below"? The first suggestion for that is in the ubiquity of the long range forces mediated by massless particles.
The sharing of subspaces is a sharing of subspaces of both carrier spaces and of the algebras. Cf. the embedding of the carrier Hilbert space into the algebra of operators below. An interlock can be characterized by the dimension of the subspace that is shared, and the number of algebras that share it.
The triangle graph is interesting because while a line segment bounded with two points is a 1-simplex, the triangle is a 2-simplex. Two form the matching algebraic expression of the 2-simplex, first remember that M(2, C) has 8 real dimensions (or 4 complex dimensions). The state space upon which the Hilbert space acts is a 2 complex dimensional vector space (a genuine Hilbert space H(2) in fact).
It is this state space that first gets connected to a line segment, by choosing some orthonormal basis of H(2) represented, with no loss of generality by
(p0| := (1, 0) and (p1| := (0, 1)
where (p0| is the complex conjugated transpose of |p0), a row instead
of a column matrix.
The normality of each of these "vectors" is expressed by
(p0||p0) = (p1||p1) = 1
and their orthogonality by
(p0||p1) = (p1||p0) = 0
which shows the notation to be essentially a "dot product" or more the
more generally referred to "inner product" of vectors. Mimicking the
style of Dirac, one can write such inner products more economically as
(p0|p0) = (p1|p1) = 1
(p0|p1) = (p1|p0) = 0
These being the basis vectors of a complex 2 dimensional vector space
and general vector can be written as a linear combination,
|s) = a0 |p0) + a1 |p1)
where a0, a1 are complex numbers. If |s) is required to be normalized,
i.e.,
(s|s) = 1,
then,
(s|s) = |a0|² + |a1|² = 1,
where |.| is the absolute value of the complex numbers, so |s) is the a convex combination of the points associated with the |p) basis vectors and so the collection of all such |s) is asociated with the line segment which connects the two points.
If we want to make the algebraic image of the same situation advanced from this 1-simplex to a 2-simplex, start with three copies of H(2), each of which has two basis vectors associated with the points at the end of a line segment. To construct the triangle from the three line segments, we'll have to make three point indentifications, which then also means identifications of basis vectors.
Let an index k = 1,2,3 label the the three line segments, and then the
basis pairs
|p0, k), |p1, k)
With perhaps over explicitness, the following identifications must be
made:
|p1, 1) = |p0, 2)
|p1, 2) = |p0, 3)
|p1, 3) = |p0, 1)
We started with the 6 endpoints, but the three identifications leaves then only three points, and those are the three points of the triangle, corresponding to the three basis vectors of a complex linear space of three dimensions, that can be spanned by three orthonormal vectors, each of which corresponds to the verticies if the triangle. The vectors of this space constrained to being convex combinations of these basis vectors then correspond to the triangle as 2-simplex. A little thought raises a question and possible objection to such an algebraic representation since it is not yet clear how this situation with an H(3) to describe a 2-simplex is different from a description of the straightline 1-dimensional configuration
o------o------o
The answer lies in differences in algebraic structure of the algebra of operators that operate on the Hilbert space H(3).
That question - and the answer is also a nonstandard answer or explanation of why of all the unitary Lie algebras, only su(2) has an IRREP for every integral dimension. We can return to that later, after the appropriate algebraic concepts have been introduced.
Before leaving that question, however, it might be worth remarking that a relevant question is, what are the ways in which an su(n) algebra can be parsed into a collection of not necessarily disjoint su(2) subalgebras that "covers" su(n)?
Here's an example using su(3). Any single su(2) subalgebra of su(3) cannot "cover" all of su(3) (say span instead of cover). If we try two, spanning takes place, but they more than share a subspace. This is the condition of the linear graph
o------o------o
Cross commutators in the Lie algebraic sense will cause the generation of a third su(2) subalgebra that closes on and completes su(3). An extended lesson from this is that su(n) algebra should be mapped to a quantum (n-1)-simplex (of n points). A point of interest is that in the n-simplex (complete graph), by permutational symmetries, the verticies are completely indistinguishable, one from another. With the previous three point linear graph, only the two endpoints can be exchanged without changing the graph. The "without changing" part is of course exactly what symmetry is all about.
Parsing su(4) into two disconnected su(2) algebras, in a direct sum corresponds to the disconnected 4 vertex graph
o------o
o------o
A relevant area of discourse is then "quantum simplicies", and wonder of wonders, these can actually be well defined.
Unitary symmetry, from the viewpoint of QT actually means the symmetry of complete interchangeability of whatever the constituants are that make the thing that is said to possess the symmetry.
If the symmetry is other than unitary it's a little less, but not necessarily such a bad thing. Specific assymmetries are also very important. Complete, absolute, maximal symmetry is a boring condition in which existence itself would be meaningless:
In Newtonian physics [w], if dp/dt = 0 expresses that background symmetry, its explicit breaking in the statement of the second law,
dp/dt = F
is profoundly more interesting and useful, in that it conveys a much greater amount of information about the structure of the physical model of reality.
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Under the rules of matrix multiplication, while multiplying
row-matrix x column-matrix
of the same dimension n, as in an Euclidean inner product, gives a
scalar, multiplying
column-matrix x row-matrix
gives what in earlier literature is called a dyadic product, but mostly
now called an outer product, that results in an nxn matrix.
M(s) := |s)(s|
maps a state |s) to a "projection operator" M(s), where a projection
operator P is defined by the properties:
P is Hermitean, i.e. = to its complex conjugate transpose,
P† = P;
P is idempotent of order 2,
P² = P;
maps the complex Hilbert space H(2) into a cone of M(2, C). (That is not
necessarily intuitively obvious, yet.) That M(s) is a projection operator
is fairly obvious:
M²(s) := |s)(s||s)(s| = |s)(s|s)(s| = |s) 1 (s| = |s)(s|
= M(s)
and by definition, (s| := |s)†, so M(s) is Hermitean.
The definition of a projection operator is well taken by its name thinking of a "projection" of a higher dimensional thing onto a lower dimensional space, i.e., the projection of a 3-sphere onto a plane is a circle. The definition says that after you have done the projection, a reprojection accomplishes nothing, i.e., the operator does not move vectors around, it merely "selects", or "projects".
If it is not true that (s|s) = 1, the M(s) might well be called a "conformal operator", since then
M²(s) = (s|s) M(s).
If and only if |s) and |t) are orthogonal, i.e., (s|t) = 0,
M(s) M(t) = M(t) M(s) = 0
or written more easily in terms of the commutator,
[M(s), M(t)] = 0.
So, if the set of an orthonormal basis {|pk)} is mapped to the set
of projection operators {M(pk)}, then for all k,j pairs
[M(pk), M(pj)] = 0,
M(pj) M(pk) = M(pk) δkj.
Then, if |s) := Σ ak |pk),
and A(s) is defined,
A(s) := Σ ak M(pk)
so,
A†(s) := Σ ak* M†(pk)
= Σ ak* M(pk)
then,
A(s) A†(s) = ΣΣ ak aj* M(pk) M†(pj)
kj
= ΣΣ ak aj* M(pk) M(pj)
kj
= Σ |ak|² = 1
k
A†(s) A(s) = ΣΣ ak* aj M†(pk) M(pj)
kj
= ΣΣ ak* aj M(pk) M(pj)
kj
= Σ |ak|² = 1
k
then, if,
M(s) := A†(s) A(s) = A(s) A†(s),
(The last equality is the definition of a "normal" operator; such animals are always diagonalizable and possess a spectral representation using an orthonornal set of eigenvectors.) [Section V] we have
M²(s) = M(s)
and so M(s) is a projection operator, directly constructed from any representation of |s) on an arbitrary orthonormal basis, consistent with the definitions of both a projection operator and with the definition of vector representation. This shows an embedding of H(n) into the C*-algebra [i]. of bounded linear operators that act on it which embedding is
|s) -> M(s)
TOC
The density matrix for an arbitrary collection of states |sk) with probabilities pk is then
M({|sk); pk}) := Σ pk M(sk)
k
is the foundational definition of quantum statistical mechanics [w] (the statistical mechanics of quantum mehanical particles) as explained by J. v. Neumann. [Neumann 1932].
Notice that M({|sk); pk}) is not a projection operator, but is a convex linear combination of them. It fails to be a projection operator because the Hilbert space states |sk) are arbitrary and not necessarily mutually orthogonal; the crossterms in the product of sums will not vanish as commutators.
Geometrically, within M(2, C), the set of all projection operators is the boundary of a "forward cone" of set of all density matrices.
The density matrices are important because they represent the most general states of patches.
TOC
Coming soon. Any year now.
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In quantum mechanics, one first encounters the infinite dimensional Hilbert space of states as the universe of discourse for any quantum mechanical system. As a consequence of the Canonical Commutation Relations (CCR), the spatial aspect of any system is infinite unless some unphysical infinite potential barrier thwarts spatial unboundedness, and yet we see with the necessarily finite nature of two point discrimination that the algebra that contains a representation of the most rudimentary aspects of space is of huge dimension, it is necessarily of finite dimension n, where n is in some sense the number of physical "points" in the universe.
However, by a well known mathematical theorem, any Hilbert space of dimension n, whether n be finite or infinite dimensional separable [w] Hilbert space, i.e., of countably infinite dimension א0, is isometrically isomorphic to any other Hilbert space of the same dimension, so in this context, the physical dimension of space is a concept that cannot be contined in, or be informed by the Hilbert space structure. That concept of spatial dimension is contained within the structure of the algebra of linear operators generated by fundamental "observables" of the system that acts on the Hilbert space. A very little thought is required to understand that the model of any "physical system" is structurally defined by the structure of its algebra of observables, which is usually generated algebraically from the elements of a Heisenberg algebra Hm, a nilpotent Lie algebra of dimension 2m+1.
In quantum mechanics, degrees of freedom specify the dimension of the nilpotent Heisenberg algebra, or perhaps of some other kinematic algebra from which the other observables are constructed.
Equivalence of multiparticle systems with one particle in higher dimensional spaces, as in classical statistical mechanics. [w]
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Addition in linearity:
Direct sum:
Direct product:
Addition in extension, e.g., extending a Heisenberg algebra by a unitary symmetry algebra
Physical irreducibility - can you pull it apart or subdivide it? reducible and irreducible representations
Example with a su(2) isotopic spin model A "nucleon" has two states: proton and neutron, being differently charged
Example with a simple su(3) quark model A "quark" has three "charge states":
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The fundamental aspect of all classical physics that is formulated in terms of differential equations is "locality", the integration of which gives the specifics of propagation arising from some set of initial conditions, or boundary conditions depending on the nature of the equation.
The tangent spaces wrongly assumed to be physically real.
Anholonomic tetrads as an expression of algebraic structure than can be given abstractly also by imposing a local Clifford structure.
gμν(x) = ½ {γμ(x), γν(x)}
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Quantum Theory interference. Electromagnetic Theory QT is a tolerance theory, and fuzzy sets, also with quantum EMT is a topological theory A, the 4-potential acts geometrically as an affine connection from which the field tensor F is derived which is its associated curvature tensor. F = dA dF = j If there exists a scalar function Φ, for which A = dΦ, then F ≡ 0. This isolates EM quantum effects from the classical effects of F since the EM field in Q theories is not F, but A, a vector which is consistent with the photon having spin 1. (The impositions of gauge inavariance and masslessness as constraints remove 2 of the four degrees of freedom in the 4-vector, leaving the two for photon polarization.) The formulation EMT in terms of differential forms on Minkowski space is formally independent of the dimension of Minkowski space. It exposes the topological nature of EMT showing that it does not depend on or require a spatial metric properties of its resident space. The exterior derivative operator 'd' has the same properties as the boundary operator of topology, and so the formal structure of EMT can be moved directly to the context of homology groups, and even more disjunctly to the context of simplicial homology where concepts of continuity are not required. Symmetries and their lack in EMT. Special Relativity SR is a (causal) theory of partial ordering by lightcones General Relativity GR is a local (causal) ordering theory plus a measure theory. This is the information content of the pseudoriemannian metric tensor field of spacetime, gμν(x), modulo the local similarity transformations induced by general coordinate transformations, which are the gauge transformations of GR. Using these gauge transformations, locally, one can always make gμν(x) look like a matrix that is the Minkowski metric multiplied by some scalar. The Einstein equations do not describe a confomally invariant theory; how that (real valued) scalar is a function over the entire GR manifold describes the measure on the manifold. The relationships among the local Minkowskian structures is the rest of the information that is not coordinate (gauge) invariant. The "Standard" Theory
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Inequivalent lattices of algebras represent inequivalent structures of universe, but representing the structure is not representing the state. The state is represented by the collection of representations of the elements of a lattice. In this way, while the lattice may "fit together nicely", as among differential structures an integrable system fits together nicely, the associated structure of representations may not fit together nicely, as analogously, an anholonomic system does not necessarily fit together nicely.
THE PARADOX: su(2) is a smallest patch necessary for high energy: you can't divide any smaller. Yet, clearly lim FCCR(n) = CCR is a limit where no energy is attributed to ST. Is su(2) highest energy or smallest?
Conceive then of a space of structured, but irreducible patches that hang together not unlike a dense gas. If one wanted to consider a neccesarily Q approach to the statistics of such an ensemble, it would probably have to be a mixture of Fermionic and Bose type objects where the patches with n even are Fermionic and patches with n odd are Bosonic. A lowest energy density in a region of patches leaves a local sea of even, n=2 patches.
Entanglements
Loop Quantum Gravity [w] Spin Networks Group + Graphs -> intertwining operators and representations. Spin Foams
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From the viewpoints of both quantum theory and relativity theory, one of the most compelling images of the BB is that of some (runaway?) quantum fluctuation that expands into this complicated existence, but, that will eventually collapse back into some unknowable nothingness.
To some, this is psychologically uncomfortable, to others of us this is comforting as almost an elegant logical necessity.
The standard terminology speaks of the begining as a "Quantum Singularity" (QS), which on the face of it is complete nonsense if one takes the singularity at all seriously. All of physics is about creating mathematical models of a physical reality that is so complex that it is utterly beyond direct comprehension, much less human experience. The idea of the "progress of physics" is simply to get a model that works and try to refine it so that it converges on a best model. History has shown repeatedly that that existing models will not suffer the appropriate refinements and that the basic model must be radically altered. There is no reason to suppose that this historical process has come to an end. [Should this analytical phase of physics conclude satisfactorily, it will not mean the death of physics: the essential rules and building blocks whatever form they may take have combined in thermodynamically reasonable ways to form structures with their own rules of combination, which in turn form higher order structures with yet a different set of rules. We will still not know what thermodynamically unlikely but stable configurations are. Even simple rules of classical automata like Conway's "game of life" have stable and isolated, stable configurations. There is yet the synthetic phase of physics to follow; with luck, when humans become much wiser than they are.]
Singularities, poles of real or complex valued functions, are mathematical aspects of those models that simply indicate that the model has failed, usually in some regime of "smallness" or "largeness". Even the ideas of actually observing or measuring singularities, or infinitesimals are equally ridiculous. Physically and realistically the BB begins with a highly energetic narrowest fluctuation. The energy of the fluctuation is not infinite, merely extremely large.
While a singularity itself is a structureless unboundedness, real and especially complex valued functions are exceedingly rich and complicated in their neighborhoods. In the case of essential singularities, one might even say infinite complicated taking as cue the Weierstrass-Casorati theorem, [w] that says of a function of a complex variable f(z) with essential singularity at z0, but is otherwise analytic in a neighborhood U of z0, that in *any* neighborhood N(z0) contained in U, there is some z in N(z0) so that f(z) is arbitrarily close to any given complex value. Better yet, the Great theorem of Picard says that f(z) in N(z0) takes on every complex value with only one possible exception, infinitely many times.
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1. Despite resorting to a good deal of specific mathematics and physics, this is very much a peroration here on "ideas" rather than any attempt to expound on a definitive theory. With the internal and external links given in absolute URLs, the entire HTML source code can easily be taken, placed and accessed from anyplace with no difficulties. The links, many of which a well schooled physicist will not need are intended for students, who should learn quickly that everything they spent all their time learning is basically wrong, but that they should learn it all anyhow in order to know from where to proceed. Note the six different types of links, other than TOC, and the links from the TOC: [XX, ddd] is a bibliographic reference to either hard copy citation or to an online paper. [w] is an appropriate external link to the Wikipedia. [i] is an appropriate internal link to some other page on this site. [r] is an appropriate link to Dave Rusin's mathematical encyclopedia. [e] is some other appropriate external link [N] where N is a number, is a footnote on this page. 2. Remark: Both trace and determinant of M(v) are invariant under similarity transformations s M(v) s⁻¹, that characterize an arbitrary change of basis. 3. This picture turns out, in part, to be very similar to Penrose's "spin networks". But, pictorially at least it appears to be dual to these since representations are attached to graph verticies while then presumably intertwining operators would be attached to graph lines. Also, here, the lines don't correspond to intertwing operators, but shared subspaces of local algebras. 4.
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