PARTS:
THE Quantum Gravity PROBLEM
AN EXTEREMLY BRIEF HISTORY OF SPACE
SPACETIME VARIABLES
QM & ITS DISCONTENTS
GR & ITS DISCONTENTS
FCCR
The magic word 'XYZZY' appears as a marker,
primarily for me, at points that need additional material
that I haven't gotten around to.
This little discourse deals in an oblique way with notions that precede those of of "Quantum Gravity". No nonexistent thing has ever generated so much talk, writing or thought. I've been thinking about it for years. Physicists, generally, have been thinking about it for about 80 years now, and nobody seems to have actually done anything crazy enough to make the whole idea sane.
I'm not really sure that my toy "solves the problem", but it looks like maybe it's a start. I have to define or better "redefine" the problem, or at least its source. This way I can indicate how the toy fits in.
My apologies if this seems rather terse and dense; it is an attempted summary of the more detailed working document of over 600 pages. I'm sure I've omitted some interesting results and some serious fretting over physical interpreatations as well as over some things that I suspect to be true but cannot prove.
As a preliminary overview (if the math & physics lingo is familiar, and even if it isn't) and a historical perspective to the following,
The Cyclotron Notebooks
by
Phillip Gibbs
is highly reccommended.
To save a few bytes, I'm going to use a set of abbreviations,
and so give the table to start with.
ABBREVIATIONS:
CAR = Canonical Anticommutation Relation
CCR = Canonical Commutation Relation
CM = Classical Mechanics (Newton)
CR = Commutation Relation
FCCR = Finite Canonical Commutation Relation
GR = General Relativity (Einstein's Field Equations)
HM = Canonically Formulated (Hamiltonian) CM
HOR = Harmonic Oscillator Representation of CCR
IRREP = IRreducible REPresentation
LM = Lagrange Formulation of CM
Q = "The Quantum Priciple"
QED = Quantum Electrodynamics
QEMT = Quantum Electromagnetic Theory
QFT = Quantum Field Theory
QM = Standard nonrelativistic quantum mechanics
QST = QuantumSpaceTime (whatever that is)
QT = Some most general quantum principle (unknown)
R = "The Relativity Priciple"
SR = Special Relativity
ST = SpaceTime
NOTATION:
The Hilbert space notation is that of Dirac.
:= borrowed from Pascal programming language is a definition:
the LHS is defined by the RHS.
^ is an exponent (chosen over the Fortran "**")
X= is not equal to.
! Hermitean conjugation
THE SOURCE OF THE PROBLEM:
Way back in kindergarten, when we first learn QM it's presented in the
Schroedinger representation as a Hilbert space (nobody says Hilbert)
of complex valued functions over the standard Newtonian space and time.
The functions must also be differentiable in their arguments and vanish
at infinity so the square of their modulus is finitely integrable over
the spatial domain. Otherwise we can't define a probability density.
Relax differentiablity but demand continuity where a potential may be
discontinuous. Otherwise we would miss out on the quantization of energy
values altogether. We learn "to quantize" by making the holy operator
substitutions in a classical expression for the energy, to wit, the
Hamiltonian function. Somehow, the ambiguity of operator ordering in
products of the canonical q, and p variables is quietly avoided.
Behold we have "derived" the Schroedinger Equation from the classical
equations of motion. Black Magic? You bet!
If we're really mindwarped by this, we eventually find Dirac's Principles
of Quantum Mechanics. Things take a slightly more elegant turn and it
is found out that Schroedinger's Representation of QM is equivalent to
Heisenberg's (which nobody told us about anyhow). Those of us with a
mathematical bent are illuninated by having the mathematical arena clearly
and elegantly defined as an infinite dimensional complex Hilbert space.
Vectors yea! Later comes the bitter truth that this is almost as bad
as the Schroedinger Stuff. No wonder, since J. v. Neumann proved a
theorem that says that any formulation of quantum mechanics is equivalent
to the Schroedinger equation. So we are stuck with the seemingly
lovely rose CCR with all its attendent thorns.
[Q, P] := QP - PQ = i h-bar I
where i = sqrt( -1 ), I is the identity operator,
and h-bar is Planck's constant divided by 2(pi).
It turns out that early papers of Wielandt, Wintner and Taussky in
the '40s proved that this expression cannot exist an any normed algebra.
Eventually the notion of "unbounded operator" comes into being.
It is mathematically necessary for at least one of the pair {P, Q}
to be unbounded for CCR to be satisfied, and it is possible to find
in the Hilbert space upon which they act a "dense common invariant
domian" for the pair, so that if we restrict the action of the pair
to this domain then CCR can be "weakly" satisfied. Symbolically,
[Q, P] |D> = i h-bar |D>
where |D> represents any vector in the restricted domain.
The universe is saved; our little "grey cells" are so completely fried /-)
having been pounced upon by unbounded operators no less!) that we would
never question CCR and its folklore of unboundedness again.
But, take a look at the HOR to be found in any decent QM text.
Messiah, e.g. Q is an infinte matrix zero everywhere but for
the lines of elements just above and just below the diagonal.
The upper offdiagonal is the same as the lower offdiagonal.
Starting from the top and counting down the k-th element of either
offdiagonal has the value
sqrt( k / 2 )
P has a similar structure, with the upper & lower diagonal elements being
respectively
-i sqrt( k / 2 ) and +i sqrt( k / 2 )
The HOR also happens to be the number representation of the creation
and annihilation operators used in QFT, and
B = (1/srqt(2)) (Q + iP)
B! = (1/srqt(2)) (Q - iP)
where ! indicates Hermitean conjugation (transpose, complex conjugate)
and inversely
Q = (1/srqt(2)) (B! + B)
P = (i/srqt(2)) (B! - B)
The number operator is defined by
N := B! B
which has an infinite integer spectral set {0, 1, 2, ...}
and the B operators obey
[B!, B] = h-bar I
There is a clear sense (that of the existence infinitesimals and
infinities) in which QM must be fundamentally wrong below some scale of
length, but its obvious quantitative and qualitative correctness above
that length makes attempting to change its foundations an operation
that is even more delicate that lovemaking among porcupines.
PURPOSE (abstract):
What I would like to show is that the traditional folklore about any
quantum theory having to be done in infinite dimensional spaces with
unbounded operators is at best highly dubious and at worst, simply wrong;
furthermore, that a good number of QM algebraic
properties can be retained by a similar model in finite dimensional
complex Hilbert spaces (finite dimensional C*-algebras) and that the
apparent restriction to finite dimensions actually allows further
algebraic detail to emerge which has the effect of:
defining a concept of *Local* *Quantum* *Theory* within which there
exists a genuine time operator not present in QM;
automatically quantizing space and time such that position and
time operators do not, in general, commute;
maintaining an uncertainty relation that is just as sensible as
that provided by QM;
providing a Fourier relation between Q and P
(In QM the technical source of the uncertainty relation);
adding an additional Fourier transform between N and a time operator T
which cannot properly exist in QM;
enriching the background structure for physical events to be a linear
space containing the concepts of position, momentum, energy and time,
which space is also provide with a natural indefinite quadratic form;
making the interesting geometrical objects (Spectra of selected operators)
quantized loop-like objects (the effective topology of the spectra is
that of a circle);
creating an algebraic bridge between CAR (n=2) and CCR (n unbounded);
showing time to be "created" in the process of algebraically necessary
transitions or "quantum jumps" of a complex Markov type process;
indicating a 'dynamics without dynamics' (in J. A. Wheeler parlance),
where the state itself does not have a determinate evolution so that
the emphasis of physical descriptor shifts from *state* to *process*
(The vectors of the Hilbert space and the density matrices of the algebra
then represent process not state);
indicating a shift of perspective from expectation values to transition
amplitudes and then transition probabilities, suggesting the
relevance of a complex or quantum Markov process as *the* dynamical
process;
not actually making the life of theorists any easier, but *possibly*
a little more sensible in terms of eliminating actual infinitesimals
and singularities, magnifying the complexity of algebraic possibilities.
The material presented here is not without problems and questions,
the answers to which may ultimately damn it to the nether regions.
It is probably incomplete from a relativistic perspective as it is
seemingly incapable, as it stands, of accomodating negative energy states,
(which may after all be introduced by a simple finite subtraction
or "renormalization"). My "feeling" at the moment is that a simple
subtraction is not sufficient.
THE "QUESTION":
What happens in the Oscillator Realization & Number Representation,
if we simply truncate the matrices to be square at some dimension n?
Writing
[Q(n), P(n)] = i h-bar G(n)
With definitions of B(n), B!(n) as above, it follows that
[B!(n), B(n)] = h-bar G(n)
N(n) := B!(n) B(n)
The Trace Tr( Q(n) P(n) ) = Tr( P(n) Q(n) ), so Tr( G(n) ) = 0,
which is clearly true for the trace of any commutator of finte matrices.
Perfoming the matrix multiplications, it isn't hard to see that G(n)
has "almost" the form of the indentity matrix: all ones down the
diagonal up to the last one, which must have the value -(n-1),
since the trace vanishes.
Since most of the proofs are long and detailed,
from here on, I'm simply going to state some results with no proof
and little, if any argument. Trust me :-).
The following hold
N(n) = Diag[0, 1, 2, ..., (n-1)]
[N(n), B(n) ] = -B(n)
[N(n), B!(n)] = +B!(n)
[N(n), B^k(n) ] = -k B^k(n)
[N(n), B!^k(n)] = +k B!^k(n)
and
[N(n), Q(n)] = -i P(n)
[N(n), P(n)] = +i Q(n)
(The carat symbol '^' indicating a power.)
It is clear that we are working in the Number Operator Representation:
all vectors are represented as linear combinations of the eigenvectors
of N(n), |n, k> or just |k> more briefly, where k = 0, 1, 2, , (n-1).
Assume the |n, k> (or just |k> when we can) constitute the "canonical
basis".
f(n) := exp( i (pi)/2 N(n) )
is a discrete Fourier transform, clearly unitary and idempotent of order 4.
Also
f(n) Q(n) f!(n) = +P(n)
f(n) P(n) f!(n) = -Q(n)
which is all algebraically *exactly* at it is with CCR.
So, f^2(n) appears to be a spatial inversion.
Complex conjugation would also appear to be a "time inversion";
this operation, however, is not invariant under general basis
changes so such appearances can be deceiving.
To see the f(n) properties, use the preceding CR's of N(n)
with Q(n) and P(n), together with the Baker-Campbell-Hausdorff
formula to show that more generally for a continuous parameter 'a',
exp( +i a N(n) ) Q(n) exp( -i a N(n) )
= + Q(n) cos a + P(n) sin a
and
exp( +i a N(n) ) P(n) exp( -i a N(n) )
= - Q(n) sin a + P(n) cos a
clearly expressing a rotation in the Q-P plane in the operator algebra.
Let a = (pi)/2 and then
f(n) Q(n) f!(n) = +P(n)
f(n) P(n) f!(n) = -Q(n)
falls out as a special case.
If 'a' varies continuously the above relations describe the behavior
of the dynamical evolution (Heisenberg picture) of the SHO in QM,
where a is then interpreted as a time parameter.
AN ASIDE:
It's always bothered me (and apparently a few other people)
that CCR (with necessarily infinite representations involving unbounded
operators) and CAR
{A!, A} := A! A + A A! = I
(with only a single 2x2 irreducible representation) should both be the
algebraic cornerstones of QM, differ only by a sign, and be apparently
unrelated and so wildly dissimilar algebraically.
In the commutation relation for the B(n), B!(n), if we take n=2, they
satisfy CAR. There is now an ALGEBRAIC BRIDGE: CAR -> CCR in the limit
as n becomes unbounded. A technical sense in which B(n) -> B is in
the strong operator topology.
I have taken to referring to this bridge as Finite Canonical Commutation
Relations or FCCR, or when necessary FCCR(n).
This bridge does not seem to have any connection to the parastatistics
that between CCR and CAR introduced by Greenberg about 20 years ago.
Nor does it seem to have any relation to the so called q-CCR which
depend on a real parameter q: -1=> q <= +1, where
QP + q PQ = I
There might be a connection with superalgebras or a generalization
thereof: both commutator and anitcommutator can be looked upon as
bilinear 2-forms on an underlying algebra.
(commutator) (anticommutator)
| 0 +1 | | 0 +1 |
| -1 0 | | +1 0 |
XYZZY
MORE PROPERTIES & DETAILS of FCCR:
Both Q(n) and P(n) are Hermitean and have real eigenvalues distributed
symmetrically about zero.
In fact, Q(n) being also real, is symmetric hence diagonalizable by a
real orthogonal transformation. (See below.) Its eigenvectors, taking
the |k> eigenbasis of N(n) to be "canonical" have real components.
The eigenvalues of Q(n) and therefore also of P(n) are the roots of the
n-th Hermite polynomial, which as n becomes unbounded become dense in
the real number field. The spectrum is nondegenerate for every n.
Only when n is odd is zero in the spectrum. An approximation for
the eigenvalues can be given:
q(n, k) = DELTAq(n) ( (n-1)/2 - k )
where the uniform eigenvalue spacing is approximated fairly well by
DELTAq(n) = sqrt( 6/(n+1) )
even for relatively small values, say 5 or 6, of n.
Although asymptotically for large n one calculates
DELTAq(n) = (pi/(2 sqrt(n))
Operator algebraic differences also occur depending on n being even and odd.
Every Hilbert(n) supports an IRREP of SU(2) and of SL(2, C).
If n is odd, these are also IRREPS of SO(3) and SO(3, 1) respectively.
The eigenvectors of Q(n) and P(n) are all G-null
<q| G(n) |q> = 0
<p| G(n) |p> = 0
Though,
[G(n), N(n)] = 0
of course, commutators of G(n) with Q(n), P(n) are not zero:
[Q(n), G(n)] X= 0
[P(n), G(n)] X= 0
These two commutators close in mutual commutation on an algebra conformal
to the SU(2) Lie algebra.
Including Q(n), P(n), G(n) in the set to be closed pushes
SU(2) -> SU(3), once again then SU(3) - SU(4) and ultimately the operators
close on and generate by commutation the full set of nxn Hermitean
matrices, i.e. SU(n). This is the cognate property of irreducibility
of the Schroedinger Operators proved by v. Neumann. (I'm told there is
now an English translation of this paper; I've not seen it.)
This implies that any operator of the full nxn matric algebra is
representable (not necessarily uniquely) as a complex polynomial
in Q(n) and P(n). From the viewpoint of SU(n), G(n) is merely an
element of the Cartan subalgebra, and not otherwise distinguished.
An interesting step that I've not yet made (takers anyone?) is an
investigation of the usefulness here of a second quantization by way
of the "quantum su(n)" group. XYZZY?
In the large and on a classical level of GR, we expect given the finite
speed of light that the seeable and knowable universe is finite, and
probably unbounded, i.e. spatially, a 3 dimensional space of roughly
constant positive curvature.
If FCCR is a legitimate usurper of CCR, n should have a finite, though
enormously large value if the universe can be almost completely
covered with one patch. How many Planck lengths do we need to measure
the radius of the known universe? On the order of 10^62. Is this a valid
application of FCCR? Maybe, maybe not.
SUPERPOSITIONS OF STATES:
The eigenstates |q> and |p> lie on the G-null cone in Hilbert(n).
Any *real* linear combination |q> is again real and also lies within
the G-null cone.
<phi| G(n) |phi> < 0
In a complex space with indefinite quadratic form, the forward
cone is continuously connected (by the phases) to the backward cone.
XYZZY (charge density?)
DIAGONALIZING TRANSFORMATIONS OF Q(n) & P(n):
The diagonalizing transformations XI(n) of Q(n) and PI(n) of P(n)
do *not* leave G(n) invariant. (Neither do general curvilinear
transformations leave the Minkowski metric invariant.)
The matrix elements of XI(n) are given by
[XI(n)](kj) = d^(-1/2)(j) H( k, q(j) )
where
d(k) := SUM(j) H( k, q(j) ) H( k, q(j) )
and
H(k, z) is the normalized k-th Hermite polynomial in z
q(j) is the j-th root of H( n, z ), where q(0) is maximal and
q(j+1) < q(j) for all j=0, 1, ..., n-1.
The cognate energy functions are <k|q(j)>.
Even for small values of n, one can see the central preaking of
<0|q(j)> that is characteristic of the Gaussian nature of
the groundstate energy function. For the "highest energy", eigenstate,
<n-1|q(j)> the distribution is essentially flat with alternating
phases of 0 and pi; the signs alternate. In a limit of unbounded n,
if one informally takes the "value at a point" to be an average over
a small neighborhood of the point, the function appraoches a zero function.
By expressing XI!(n) XI(n) = I(n) in components, one obtains an
orthogonality relation for the Hermite polynomials on a discrete set.
SUM(i) H(k, q(i)) H(j, q(i)) = delta(kj)
objects being defined as above.
I suspect that this is a special case of a general theorem valid for
a certain proper subclass of orthogonal polynomials, but I've neither seen
the more general theorem, nor stated it precisely, much less proved it.
SOME MATRIX VALUES:
<q(k)| P(n) |q(j)> = 0, k = j
= -i exp( i(pi)(k-j) )/( q(k) - q(j) ), k X= j
<q(k)| G(n) |q(j)> = kronecker-delta(k, j) - exp( i(pi)(k-j) )
UNCERTAINTIES:
The proof of the uncetainty relation goes through in QM textbook fashion.
[See e.g. Merzbacher's "Quantum Mechanics"]
Interestingly, it has the same point of failure as exists in QM.
In slapping a bra on the left and the conjugate ket on the right of CCR,
these must not be eigenvectors of either Q or P.
In QM, this case fails because these eigenvectors happen to lie outside
the common domian of selfadjointness for the operator pair.
In FCCR, the proof fails because all these eigenvectors are G-null,
making the RHS of FCCR statement vanish. If the uncertainty in Q(n)
vanishes, there is still an uncertainty in P(n) which is very approximately
25% of the full spectral diameter of P(n); the percentage is independent of n.
Gratuitously said, the uncertainty in P(n) in this case become infinite
when n becomes infinite.
In QM, we have had to consider certain states to be unphysical,
particularly the eigenstates of Q and P. They are not strictly
elements of Hilbert space and not strictly normalizable. The
Dirac delta-function is not strictly a function. Unlike a general Banach
space a Hilbert space is automorphically equivalent to its space of
linear functionals. To make mathematical sense of manipulation of
these useful eigenstates it becomes necessary to invent the theory of
distributions and split the Hilbert space
into a smaller space (a space of test functions) and a larger space
of distributions. Symbolically,
test functions < H < distributions
Then a expression (test-f)*(distib) adequately replaces the expression
fg, with f and g both in H.
In FCCR one must either do something similar, or accept that the vectors
|psi> for which
<psi| G(n) |psi> = 0
actually have physcial significance.
There is a definite problem with the first choice: Since by commuation
FCCR(n) generates the full SU(n) algebra defining IRREP, there appears to
be no way of consistently eliminating vectors of the Hilbert space and
defining an *invariant* and proper subdomain. It would then appear that one
must take the G-null states seriously from a physical viewpoint.
We have had experience with a structure that instrinsically incorporates
similar null-vectors on a cone and that structure is SR.
The suggestion is immediate that the G-null vectors of FCCR be associated
with "the local light cone". The Q(n) and P(n) eigenvectors should then
said to be on the light cone. Since they clearly involve a sense of time
they must be understood as representing aspects of *lightlike* *process*
rather than that of "state".
The eigenvalues of Q(n) may be considered to be either measures of
length or measures of time and likewise the eigenvalues of P(n) may
be considered to be either measures of momentum or measures of Energy.
Linear combinations of G-null vectors are generally not G-null.
XYZZY
THE TIME OPERATOR:
In QM, for the SHO or the creation and annihilation operators, there have
been numerous attempts at defining a "time operator"; this should be
associated with a phase operator (Hamilton-Jacobi theory) conjugate to
the number operator. The problem is that strictly speaking (the
unboundedness problem again) the phase operator doesn't exist.
An unbounded operator can be recognized by an unbounded spectrum.
Clearly the number operator N of CCR is unbounded.
When n is finite, however, one can decompose B(n) and B!(n) as the
product of an Hermitean operator and a unitary operator thus:
B(n) = C(n) N^(1/2)(n)
B!(n) = N^(1/2)(n) C!(n)
where C(n) is the cyclic operator in the eigenbasis of N(n) such that
C(n) |k> = |k-1>
where
N(n) |k> = k |k>
k = 0, 1, 2, ..., (n-1)
and the index of the basis set k is understood cyclically, that is
|n-k> = |k>
Now since C(n) is unitary and idempotent of order n, it has as
eigenvalues the complex n-th roots of unity, and has a generator T(n),
so it can be written
C(n) = exp( i 2(pi)/n t(n) T(n) )
or
C(n) = exp( i w(n) T(n) )
where w(n) is a parameter depending on n and having the physical
dimension of (1/TIME) and T(n) is an operator with eigenvalues
t k
where t is some "fundamental time quantum" and k is an integer.
Then T(n) is unitarily equivalent to (t N(n)), and there is a
a unitary diagonalizing transformation for T(n)
U(n) T(n) U!(n) = t N(n)
which also happens to be a Fourier transform, that relates N(n)
and T(n). U(n) is unitary and idempotent of order 4.
Its matrix elements can be given by
<k| U(n) |j> = sqrt( 1/n ) exp( i (2pi/n) kj )
The eigenvectors of T(n) can be expressed as *equally* *weighted*
superpositions of the N(n) states |k> states, which represent the
pointer positions of an optimal or best clock.
See [Wigner Rev. Mod. Phys. 29, 255-268 (1957)] for such a construction
in the context of QM.
Actually having a "time operator" operator, however is
like the dog chasing the volkwagen actually catching it. Now what
do we do? The position operator Q(n) does not commute with T(n).
In fact T(n) doesn't seem to commute with anything. The commutators
[N(n), T(n)], [G(n), T(n)]
don't have any easy algebraic relation and the first one is definitely
not equal to iG(n). The fact that G(n) seems to become an indefinite
bilinear form on the finite dimensional Hilbert space that one may have
to take seriously as an inner product, appears troublesome.
It is true that if n is even,
[f^2(n), U^2(n)] = 0
How might it be possible to interpret this algebra relying on what's
come and gone before.
On G(n) not being the identity operator.
In QM one talks about measurements of Q and P at the same time
being incompatible, the incompatibility being quantified by CCR
and Planck's constant. FCCR says "You can't talk about these
measurements being at the same time." So G(n) must have something
to say about time measurements.
A major change in thinking about NM happened with the introduction
of HM. Prior to HM, the "background" of physical events was space.
In canonical formalism, the background of physical events becomes
the phase space with a metric derived from the Hamiltomian energy
function. This loosening of the background concept is enhanced &
reinforced by QM where the phase space is replaced by a projective
Hilbert space.
SR loosens the background concept in a completely different direction
leaving the absolute space and absolute time behind for the looser
concept of and absolute spacetime. GR further pulls stress, energy,
and momentum together into one tensorial object.
According to FCCR, it would seem that position, time, momentum and
energy are all "Q-incompatible" and one can chose only one to
diagonalize. If we take G(n) to be "the metric" by which one
computes probabilities, negative probabilities will result.
It may be proper to take G(n) as an operator invariant under a
group of symmetry transformations, but it is not proper to replace
<psi|psi> with <psi|G(n)|psi> and "normalize a probability density"
by it. Once can, of course, fix the value of
<psi|G(n)|psi> = +1
in imitation of the QM, but then all the |psi> conforming to this
constraint lie on a hyperbolic complex manifold embedded in
Hilbert(n). This hyperbolic manifold is not closed under linear
combinations (neither is the G-null cone) and one is then stuck
with relinquishing the superposition property, and possibly finding
some other nonlinear rule for the combination of vectors in
Hilbert(n). This sounds like an interesting line to investigate, but
I haven't gotten around to it.
An alternative to holding on to the QM interpretation of the mathematical
formalism is to abandon the dominance of probabilities and
allow that the fundamental computable numbers of physical interest
are *Transition amplitudes* (shades of Feynman path integrals), and
not *expectation* *values*.
Concomitantly, the fundamental physical concept ceases to be that
of "quantum state" and become that of "quantum process".
The consideration of process rather than state is already made
necessary by T(n) not commuting with anything.
If G(n) is construed as a metric on the background which is the
local finite dimensional C*-algebra, it is a metric on a space
which unifies:
P(n) E(n)
Q(n) T(n)
moreover it is an indefinite metric not unlike the indefinite
metric on (x, t) space [contravariant] and on (p, E) [covariant]
put together in one space. The conclusion is that FCCR may
provide a correct, if somewhat wierd, coalescence of R and Q
that is consistant.
The group of transformations that preserve G(n) is conjugate
equivalent in GL(n, C) to SU(n-1, 1), and of course, has a
maximal compact subgroup SU(n-1). When n=2, it is exactly
SU(1, 1), the isotropy or "little" subgroup of SL(2, C) which
leaves invariant a spacelike vector.
THE REAL 'STATE' IS THERMODYNAMIC & IN THE ALGEBRA, NOT THE HILBERT SPACE:
A local FCCR algebra pertains only to a patch of the universe;
A QT of a local *open* system which presumably is in thermal
contact with "the rest of the universe" as a thermodynamic
reservoir. This is to say that a local "state" or process is
entangled with its other, and that a realistically considered
process is necessarily impure. The impure processes should then
be represented by density matrices. To decompose such density
matrices into a probabilistic distribution (in the classical sense)
of pure states is not uniquely defined.
The Hilbert space can be absorbed into the algebra in the usual
way by the map |psi> -> |psi><psi| to projection operators which
will be extremal in the forward cone of density matrices *if*
the inner product on the algebra is given by Tr( Y Z ) with Y, Z
elements of the algebra. No details on the alternative inner
product Tr( Y G(n) Z ) are given here.
DYNAMICS WITHOUT DYNAMICS:
The "state" does not have a determinate evolution.
It evolves statistically by transition amplitudes as Markovian
process. It is this process which appears to create time.
If one looks at the values
|<qk|tj>|^2
numerically, they appear to have maximal values that describe
a sinusoidal curve. I have tried to prove this analytically
and not been successful. It is not hard to see that these transition
probabilities form a doubly stochastic matrix.
If one looks at the values of <tk| E(n) |tj> where
E(n) = (1/2)( Q^2(n) + P^2(n) ) Oscillator energy
= N(n) + (1/2)G(n)
taking still as axiomatic that energy is the generator time
translations, then for n=2 and n=3 the "clock pointer positions"
evolve chaotically, for n > 3 the transitions tk -> t(k-1) and
tk -> t(k+1) dominate. As n increases further these adjacent
transitions dominate even more, so that as n inceases the
n-point oscillator behaves more and more like a smoothly running
clock. Remember that unlike QM, the energy eigenstates are *not*
stationary, since the energy does not commute with the T(n) operator.
There is another aspect of the transition amplitudes for the
n-point SHO that is important. The transition amplitude for
no transition (expectation values) does not vanish, so there
is an intrinsic speed of the clock that can be statistically
calculated. A rate of transition is determined also not just
the expected rate of the clock. Time would then seem to appear
almost magically as a consistantly determined effect, independent
of any prior measure except that of a given quantum.
An observed speed of the clock can slow if the change of reference
frame increases relatively the probability of stasis.
Darwinism and the essential "randomizer".
XYZZY
MULTIPLE CONCEPTS and FORMAL TIME STRUCTURES:
XYZZY
THE DIMENSION OF ST, QST & ALL THAT:
Fixed v. Floating Dimension.
Various definitions of "dimension"
pre/times
Primacy of Lightcone (G-nullity) & Penrose Twistors
construction of superalgebra and quantum group
XYZZY
EXTENDING TO 3-DIMENSIONAL "SPACE"
The identity and integrity of a quantum system is recognized
by irreducibility of its maximal Abelian operator set acting
on its space of states.
The rule for combining systems in QM is the direct product.
It is also the usual rule for extending spatial dimensions.
All infinite dimensional separable Hilbert spaces are essentially
equivalent, what distinguishes them in QM is the structure of
the set of operators that we set up as the "observables of the
system". All finite dimensional Hilbert spaces are not essentially
equivalent; they are partitioned into equivalence classes labeled
by their topological invariant dimension. The size of a "Qset"
measured by the n of an Hilbert(n) is distinguished from the
dimension of the Qset that is determined by the algebraic structure
of the fundamental operator set associated with the definition of
a physical system. It would seem appropriate to seek the extension
of spatial dimension in the extension of the operator set rather
than resorting to extensions by the direct product operation.
The Q(n)-P(n) pair or the B(n)-B!(n) pair already generate by
commutation, the full SU(n) algebra (in defining representation)
space over the reals, and then that of SL(n, C) by complex extension.
If one extends the FCCR(n) set to include other Q-P pairs then
the different Q's and the different P' will not commute among
themselves, producing and explict model of a noncommutative
geometry.
Let
W(n) := (n-1)I(n) - N(n)
(which looks like a number operator turned upside down).
Let
S(1, n) := (1/2)( B!(n) W^(1/2)(n) + W^(1/2)(n) B(n) )
S(2, n) := (i/2)( B!(n) W^(1/2)(n) - W^(1/2)(n) B(n) )
S(3, n) := (1/2)( W(n) - N(n) )
then
[S(a, n), S(b, n)] = i epsilon(a, b, c) S(c, n)
where a,b,c = 1,2,3 expresses the CR's for SU(2) algebra.
In the basis |q> in which Q(n) is diagonal define an SU(2) IRREP
with the same numerical values as the S(a, n) preceding, calling
the operators SQ(a, n) so that
[Q(n), S(3, n)] = 0
exponentiate the SQ's for the rotation operators
R(1,3; n) := exp( i (pi/2) SQ(2, n) )
R(2,3; n) := exp( i (pi/2) SQ(1, n) )
[don't worry about the sign ambiguities when n is even.]
and let
Q(1, n) := R(1,3; n) Q(n) R!(1,3; n)
Q(2, n) := R!(2,3; n) Q(n) R(2,3; n)
Q(3, n) := Q(n)
which are to be taken as a set 3 noncommuting position operators.
The differences between successive eigenvalues of the SQ(a, n) is
exactly 1, uniformly (excepting at a cyclic wraparound. The same
differences for the Q(a, n) are asymptotically and uniformly
DELTAq(n) = sqrt( (pi/2n) ), which is to say that the Q(a, n) are
conformal to the SQ(a, n) and that asymptocially one has
[Q(a, n), Q(b, n)] =~ i DELTAq(n) epsilon(a, b, c) Q(c, n)
as n goes to infinity,
[Q(a, n), Q(b, n)] -> 0
Now make a similar construction in the |p> eigenbasis with giving
a new IRREP of SU(2): SP(a, n) perform similar rotations on P(n)
to define P(a, n) which will by the same reasoning obey
[P(a, n), P(b, n)] =~ i DELTAp(n) epsilon(a, b, c) P(c, n)
We also happen to know that DELTAp(n) = DELTAq(n), and that
as n goes to infinity,
[P(a, n), P(b, n)] -> 0
Then define 3-space operator metric G(a,b; n) by
[Q(a, n), P(b, n)] = i G(a,b; n)
But
SIGMA Q^2(a, n) = SIGMA P^2(a, n) = (n^2 - 1)/4 I(n)
showing the Q's and P's as operator 3-vectors to be on a "quantum
sphere" of asymptotic radius (n/2)
Additionally, SQ(a, n) is unitarily equivalent to SP(a, n) and so
there exists a V(n) with V(n) V!(n) = I(n) which maps one IRREP
into the other and then asymptotically maps
V(n) Q(a, n) V!(n) = P(a, n)
Is it necessarily a Fourier transform?
It is not difficult to show that
V(n) = XI!(n) PI(n) = XI!(n) f(n) XI(n) f!(n)
and
V!(n) = f(n) XI!(n) f!(n) XI(n)
V^2(n) = XI!(n) f(n) XI(n) f!(n) XI!(n) f(n) XI(n) f!(n)
XYZZY
If one attempts to make a simple minded SHO by summing over the
E(n) = SIGMA (1/2)( Q^2(a, n) + P^2(a, n) )
for the three values of 'a', the energy becomes completely degenerate
(this is the sum of Casimir operators for two IRREPS) with
E(n) = 2 (n^2 - 1)/4 = (n^2 - 1)/2
A proper energy for such a system would more likely be some operator
quadratic form such as
E(n) = SIGMA G(a,b; n) Q(a, n) Q(b, n) +
SIGMA G(a,b; n) P(a, n) P(b, n)
modulo the usual problem of operator ordering. I have not yet looked
into this structurally. Here again the thought pattern is definitely
nonrelativistic. There is no guarantee that any quantum behavior has
a classical limit, but if it (is supposed to, etc, ...) then the
standard prescription of symmetrizing of all possible orderings may
be appropriate.
This is all just model playing for the purpose of demonstrating that
such games can be played with. Since we are finite dimensional spaces
there is room for serious numerical playing around. To generate
simple position operators Q(a, n), I've simply used what amounts to
a property of SU(2) and a tacit assumption of isotropy, i.e. a
local symmetry that appears to be not unreasonable.
Just the idea of symmetries expressed by the algebra attached to some
Lie group is probably a good start for playing.
Some metaplaying: In CM the thing to define to characterize a system
is the energy or Hamiltonian as function defined on phase space.
Why? Because almost all the essential laws of physics are basically
statements of the conservation of energy E (or some loosening
generalization thereof). In SR particle mechanics, E becomes merely
a component of a 4-vector and it is (E^2 - p^2) that is conserved
under the group of Lorentz transformations (Poincare actually) and
even the conformal group, ..., blah, blah. Relativistic systems
with interactions are hard to deal with. Try making really good
sense with the angular momentum of the electromagnetic field.
SR implies the necessity of fields for the interactions precisely
because it denies action at a distance (Hoyle & Narlikar notwithstanding).
Hamiltonian formalism which works so elegantly in CM becomes
problematic in SR systems: 'Frozen Formalism' [See e.g. A. Mercier,
"Analytical and Canonical Formalism in Physics", Dover Pubs.] - A
brief but enlightening book. XYZZY
In (x^ - c^2 t^2), transforming the x away by a Lorentz boost to
the comoving frame. If there is some time operator T(n) gotten by
magikal incantation (no energy operator having been defined), then
there should be (so say I) an action of a boost transformation
(hyperbolic rotation):
"inner action" "outer action"
A(theta) Q A!(theta) = (cosh theta) Q(n) + (sinh theta) T(n) = Q'(n)
A(theta) T A!(theta) = (sinh theta) Q(n) + (cosh theta) T(n) = T'(n)
XYZZY
GR equates two forms of energy
[Einstein tensor] [Stress-energy-momentum tensor]
G(mu, nu) = k T(mu, nu)
where LHS is the Einstein Tensor expressing "energy" in terms of
geometrical curvature and the RHS is the k the gravitational
constant times the the stress-energy-momentum tensor expressing
energy in terms of matter, fields and whatever is not strictly
geometry. To build a cognate of the Riemann curvature tensor,
or even the Christoffel symbols of affine connection from a metric
field it is necessary to be able to take a "derivative with respect
to coordinate". But the coordinates here are operators. I have
been thinking about definitions of operations that take the appropriate
kind of derivative but am not ready to speak yet. Nevertheless, under
the restriction to a space of constant curvature, the Riemann tensor
becomes expressible purely algebraically in terms of the metric
(equivalent to a metric induced on bivectors). Classically it is
written
XYZZY
A MODEL OF STATISTCALLY DETERMINED DIMENSION:
In any Q theory in an algebraic context it is difficult to get
away from the notion of some fixed dimension for space time that
is basically the standard axiomatized Brouwer-Uhryson dimension.
Somehow it always seems to be put in, sometimes covertly, "by hand".
Yet there are models that can be built where a dimension (not
necessarily integer valued) is determined "thermodynamically"
by a statistics of graphs that are formed by a set of verticies
that may or not be connected by lines all of the same (interaction)
length. The idea starts by defining the dimension at a vertex
for such graphs as the number of connecting lines. A fully
connected set of (n+1) verticies is an n-simplex; to each of the
n+1 verticies n lines are connected. This (classical) model
envisions the universe as essentially crystalline in nature and
appropriately predicts a very low temperature for the current
universe. I have tried to see a connection between FCCR(n) and
such graph statistics but have not been successful.
Either this is nonsense or I haven't thought long or deeply enough.
Definitely a topic for the future.
CAVEATS ON APPARENT SIMPLICITY:
Replication, combininig, Interaction, Linking are going to be messy.
Harmonicity is not a relativistically invariant concept (except on the
light cone), this, despite many people writing down what would appear
to be relativistic equations for a simple harmonic oscillator.
The seas are not necessarily simple because they are "merely"
H-O-H. I envision FCCR as being the metaphorical cognate of
of a water molecule. Higher structures must arise from the ways
in which replications assemble themselves.
THE 'IFS' OF FCCR:
Either FCCR(2) is the extent of the cosmological initial condition,
or the initial condition is specified by some ensemble of them.
Then, why not an ensemble of FCCR(k) for all k < n, for some n.
XYZZY
ON PUTTING LOCAL ALGEBRAS TOGETHER & THE QST CONCEPT:
The general QT method of combining systems is direct product.
FCCR(k) generates by commutation, the defining IRREP of the Lie
algebra SU(k). For fixed k, the direct product of m such IRREPS
decomposes by "mj-symbols" into a direct sum of IRREPS of SU(k).
The direct product
SU(n) X SU(m) is a subalgebra of SU(nm) algebra.
XYZZY: for k>2 SU(k) has k-1 "Fundamental Representations" that are
*not* unitarily equivalent. SU(2) is distinguished by having
only 2-1=1 such fundamental representation.
SU(3) has two fundamental representations (usually denoted as
3 and 3*.
The SU(k) Lie algebras are the only ones that have the property
that the fundamental IRREP direct product
[adjoint IRREP] [singleton IRREP]
k X k* = (k^2 - 1) + 1
The adjoint IRREP (also called the vector representation)
of any Lie algebra is the representation constructed from
the structure constants.
In QM one sees space emerge from the spectra of position operators.
Each eigenvector |q> of Q is associated with a point; never mind
that the vector is not strictly in the Hilbert space; any vector
of the projective Hilbert space can be represented by a linear
combination of the |q>'s. One might think that garden variety QM
is already telling that there is already something wrong with "point".
It is possible, of course, to interpret the failure of the proof
in QM to be nothing more than a mathematical inconvenience; however,
failure, even in the very similar mathematical model FCCR, might
suggest that one take the failure a bit more seriously in terms of
actual physcial significance.
Continue this association nonetheless. The smallest set of quantum
points contains at least two points (n=2) at a maximal separating
distance. As n increases, the Spectral radius of Q(n) increases as
sqrt( n ) and the distance btween the eigenvalues of Q(n) decrease
as sqrt( 1/n ). More allowed space implies more allowed energy and
with higher and higher energies the resolution of points becomes
better.
If we want to link local algebras together to form a universe,
first think in terms of the Hilbert spaces, which should be linked
by sharing subspaces: the quantum equivalent of set theoretic
intersection. Then the power set or set of all subsets of nonquantal
points becomes the complex Clifford algebra of all subspaces of the
Hilbert space. The C*-algebra then being the direct product of
Hilbert space with its conjugate space, we must take the direct
product of the Clifford algebra with its conjugate to express the
cognate quantum set theory. An obvious question is one of the
statistics of quantum points. Do they have Fermionic or Bosonic
properties, or both? Are "quantum points" the ultimate dynamical
objects? The pertinant objects here seem rather to be the mixed
background sets that constitute the local algebras. Space, Time,
Momentum and Energy become tangled up, which is actually proper if
one considers the flat spacetime that we see locally to be itself
a statistical average of of some underlying quantum dynamics.
In this view, the "presubsets" of spacetime correspond to the local
C*-algebras: subalgebra -> subset. When n is even half-odd integral
spin (fermion) IRREPS of SU(2) are supported, and when n is odd,
integral spin (boson) IRREPS of SO(3) are supported. There is no
other unitary group that has exactly one IRREP for for every n.
All the finite dimensional (nonunitary) IRREPS of SL(2, C) can be
obtained as operators of homogeneous polynomials of two complex
variables (or essentially putting together two IRREPS of SU(2).
It might be correct therefore, to consider a combined (direct product?)
pair of FCCR's instread of just one.
A MORE GENERAL PASSAGE: LOCAL -> GLOBAL
C*-bundles
C*-algebraic presheaves
XYZZY
PROBLEM 1:
Untangle the multiplicity of time concepts.
PROBLEM 2:
The dimension problem.
There seem to be (only?) two approaches to an explanation of
of why spacetime has on the average (in the semilarge) its
manifest 3+1 structure. The first is to explain it from a
highly peaked average of underlying statistical events.
This is in accord with our picture of the flat Minkowski
space representing a local notion of vacuum, and a violence of
its attendant quantum fluctuations. The second is to understand
dimension as a necessary algebraic constraint. Although, to me
personally, the first, statistical approach is more conducive,
FCCR would seem to lean in favor of the algebraic constraint.
The question, in the this second case would still remain: why
the algebraic contraint? Are we not simply putting the
phenomenology of classical dimension back in by hand?
PROBLEM 3:
Develop an local addition on algebras attached
Presheaves of finite dimensional C*-algebras.
PROBLEM 4:
Since the rotated Q(n) operators are asymptotically
conformal to an n dimensional IRREP of SU(n) is there a
connection here with the machinery of "spin networks"
introduced by Roger Penrose? The connection being in
the interconnections of local algebras.
PROBLEM 5:
Define Field operators on such a local QST patch as this.
PROBLEM 6:
Explain how the "expansion of the universe" comes about.
The process should apparently involve an evolution of the
sort
FCCR(n) -> FCCR(n+1)
mirrored in
SU(n) -> SU(n+1)
REMARKS:
Although this may all seem a bit crazy, but I see some virtue in
eliminating from physical theory, nasty and unobservable things
like infinitessimals and singularities. For large n, behavior
approximates QM, and for any finite n, however large, there are
no singularities. If sanity is the criterion, consider the sometimes
crazy consequences of QM, especially with Copenhagen interpretation
appended.
For years people have attempted to find some picture of spin by
using some analog of classical angular momentum. All such attempts
that I've seen are unstatisfactory and somewhat tortured messes
leaving one just as geometrically confused as before. Although
this enterprise seems reasonable and natural enough at first,
perhaps a better geometrical understanding of spin might be had
by considering it as an aspect of harmonic oscillator of the type
presented here.
There is logical virtue in "explaining CCR" and therefore QM in
terms of a limite of finite algebra. QM is normally arrived at
through the rather perverse (ill defined, and backward) process of
"quantization" that is supposed to map a classical system to a
quantum system. The logical perversity here is the underlying
presumption that somehow the classical picture is primary and that
QT is merely an inconvenient afterthought. Without question the
shoe is on the wrong foot. Starting with an already quantized
everything is logically preferable to forcing classical mechanics
on a QT, whatever form that might ultimately take.
Presented with the infinitely many
degrees of freedom in CM it is quite impossible to say which of
all mathematically possible systems indeed exist physicallly.
We basically make models up that seem to fit the classical
phenomenology. If, on the other hand, one starts with some axiomatic
scheme with a finite number of degrees of freedom, the separation,
if indeed a separation is necessary, of the physically possible
(and therefore compulsory) systems from the impossible systems
seems to become tractable as a problem or statement.
Any comments, connections, remarks or constructive criticism :-),
gratefully accepted and acknowledged.
---------------------------------------------------------------------------
A Brief History (M. Jammer)
Greeks
Primitive Continuum of Space
Atomists: Democritus, Leucippus
Paradoxes - Zeno
The Matter-Space dichotomy mirroring the Mind-Matter dichotomy.
The Continuum Formalized
Dimension as a toplogical invariant and alternative definitions (Hausdorff)
Peano space filling curves.
Concept of "Dynamical Dimension" from a statistical graph model.
Space and time concepts to be distinguished
Phenomenological
Operational
Formal Mathematics
Clifford-Riemann-Einstein Program of geometrization.
Classical extension of "the background" space -> phase space
Hamiltonian Mechanics.
SR extension of the absolute space & absolute time to a looser
"absolute spacetime".
SR is not about "relativism" but about absolutism and absolute
invariants under a group of transformations, and has therefore
a most unfortunate monicker with which we are stuck.
Adding SR + Hamiltonian -> Frozen formalism
Space Time
Momentum Energy
Paradox of time: we use it to measure it.
What's going on here?
The paradox of space measured by rulers and clocks.
The assumption (and it should be a valid one) must be made
that local rulers and clocks of some sort exist.
Periodic local structures.
Rosenfeld and the point concept in QEMT
The formal correction of QM to agree with SR and then with GR is a disaster.
The parallel of Einstein correcting CM to agree with EMT.
Which is wrong?
A physical Dubiousness: SR in GR is in tangent and cotangent spaces
"at a point!", not actually "in" the spacetime manifold.
Under a condition of high curvature at P, the tangent space is not
a very good approximation.
So what does it mean to say "GR approximates SR 'locally'"?
The only really good physical 'locally' is at a point -
but Rosenfeld ....
The ambiguity of quantization as a general map.
The mutual contradictions of R and Q:
Q is an operator, t is a parameter
R indicates an absolute determinism,
Q on the other hand implies an indeterminism of both
past and future: by the "time" I hear you say
"now", "now" is already past and our illusory
notion of time doesn't really exist.
When does the indeterminism become classically determinate?
(The "collapse of the wave function" problem.)
Although in the context of QM one can say many experimentally
sensible things, one can also say many crazy things that are not
even really crazy enough. The artificial and unnecessarily inserted
observer implicated in most texts on QM is clear nonsense.
Einstein's question to Bohr: "Do you really believe that if
I don't look at the sun it doesn't exist?"
Then there's all this psychophysical parallelism (v. Neumann) stuff.
There has to be a middle ground which means one cannot correct
R as formalized to agree with Q, nor the other way around.
Both the Q principle and the R principle must currently be formalized badly,
or too specifically formalized. FCCR loosens this specificity yet
continues the results and consequences of CCR that are necessary
for essential physical correctness of QT as presently understood.
It is clear (to me at least) that the entire *formal* business
as currently set up for both R and Q is patent nonsense,
and that something truly insane is necessary to make things sane.
Probably a mathematics that can accomodate both a local C*-algebraic
structure as well as a manifold-like structure.
The two immediately obvious candidates are manifolds with complex
or almost complex structures, and C*-bundles where the fibres are
C*-algebras and the cross sections of the bundle have to do with
spacetime structure.
One can think formally of QM and QFT in terms of C*-bundles where
the base space is a classical (Newtonian or Minkowskian) spacetime
and the fibres are C*-algebras.
Unfortunately, both of these nifty if rather sophisticated structures
bring one back immediately to the problem of the single point.
The linear complex spaces (tangent spaces) of the manifold are
not in the manifold and locality once again is formally only a point.
With a C*-bundle one is really back with a picture of a "standard"
QFT: the objects of interest are operator valued functions defined
on some *continuous* space.
No one has ever seen nor measured directly or indirectly an infinity
or an infinitesimal; it seems most unlikely that anyone ever will.
On a fundamental level, spacetime must be discrete but clearly not
in a classical but quantum sense.
A proper foundation of physics must begin formally with finite algebra
(it looks like Lie algebras) and counting.
Darwinism and the randomizer.
Natura salutus facit sempre - the source of the randomizing
QM = CM + CCR
Outline of the messiness of CCR
No time operator.
Hilbert Folklore of QM.
Theorems of Wielandt & ?
Finite dimensions are out.
Since in their proper regimes both QM and most particularly QED
have amazing success, more than one could hope for from such
formal nonsense, try to take the essence of QM call it QT and
the essence of the "Relativity Principle", the essence of ST
and form a logically consistent theory.
There, that ought to about do it.
[And now I sit down :-)]
FCCR
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