Limits of the phase operator and its canonical commutator
In the limit as n->infinity, for k, j = 0, 1, ..., n-1, the variable
omega := pi (k-j)/n (17.1)
becomes a continuous variable with range - pi <= omega <= + pi.
In the limit as n->infinity, for m = 0, 1, ..., n-1, the variable
s := m / sqrt(n) (17.1)
becomes a continuous variable with range 0 <= s < infinity.
The limit of the unitarity expression for the Fourier transform
UPSILON(n) defined by (7.21):
(n-1)
(1/n) SIGMA exp[ i2 pi(k/ sqrt(n) - j/ sqrt(n))(m/ sqrt(n)) ] ->
m=0
infinity
(1/(2 pi)) INTEGRAL exp(i 2 pi(x - y)s) d(2 pi s) (17.2)
0
which defines a Heisenberg delta distribution.
infinity
INTEGRAL exp(i 2 pi(x - y)s) d(2 pi s) = delta_+(x - y) (17.3)
0
The unitarity of UPSILON(n) is metaphorically cut in half.
Considering also the limit of the phase operator F(n) given by
F(n) = UPSILON(n) N(n) UPSILON!(n)
<n, k|F(n)|n, j> =
(17.4)
(n-1)
SIGMA (m/sqrt(n)) exp[ i 2 pi(k/sqrt(n) - j/sqrt(n))(m/sqrt(n)) ] (1/sqrt(n))
m=0
we have with alpha = lim k/sqrt(n), beta = lim j/sqrt(n), s = lim m/sqrt(n)
infinity
<n, k|F(n)|n, j> -> INTEGRAL s exp(i 2 pi(alpha - beta)s) ds
0
Now
delta_+(2 pi beta) = (1/2 pi)delta_+(beta) (17.5)
and
(PARTIAL/PARTIAL beta) delta_+(2 pi beta) =
infinity
(i 2 pi) INTEGRAL s exp(i 2 pi beta s) ds (17.6)
0
The distribution delta_+( beta ) has the property
beta (PARTIAL/PARTIAL beta)delta_+(beta) = - delta_+(beta) (17.7)
So in the limit
i delta_+(alpha - beta)
<alpha|F(n)|beta> = ---------- ---------------------
(2 pi)^(2) (alpha - beta)
-i
= ---------- delta^(1+)(alpha - beta) (17.8)
(2 pi)^(2)
and
i
<alpha|[N, F]|beta> = ---------- delta_+(alpha - beta)
(2 pi)^(2)
(17.9)
where delta^(1+)(alpha - beta) is a first derivative, and formally,
N := lim (N(n) / sqrt(n)) (17.10)
n->infinity
with a continuous spectrum bounded below. This limit then
is not that of the QM harmonic oscillator.
In QM, when the position operator is in diagonal form and treated as a "multiplication operator", the momentum operator is represented as a derivative operator. We say this as simple hand waving, and do not bother here about the messy details of distributions in defining what it means for a square integrable SCRIPT-L^(2) function to be differentiable. In fact, it is not so odd that P(n), or Q(n) can behave like a derivative since mappings A -> [A, M] are derivations in matrix algebras and obey non-commutative versions of all the Leibnitz formulas. [Putnam 1951] and [Appendix B].
The Fourier transform has been used to extend the notion of derivative [Dold 1975], and can also be used here in its discrete incarnation to define a notion of discrete derivative.
In the |q(n, k)> in which Q(n) is diagonal, Q(n) acts like a multiplication operator. Cf. [Section IX].
Let psi(q) := <psi|q>, (q = q(n, m)) in Sp( Q(n) )
be a function defined on the spectrum Q(n).
(Q(n) psi)(q) = <psi|Q(n)|q>
= q <psi|q>
= q psi(q) (17.11)
and
(P(n) psi)(q) = <psi|P(n)|q>
= SIGMA_(jk) <psi|q(n, j)><q(n, j)|P(n)|q(n, k)><q(n, k)|q>
= SIGMA_j <psi|q(n, j)><q(n, j)|P(n)|q>
exp(i pi (j-m))
= -i SIGMA psi( q(n, j) ) ----------------- (17.12)
(q(n,j) not= q) q(n, j) - q(n, m)
which has the formal appearance of a Hilbert integral transform
of a complex valued function made discrete, or a discrete version
of an integral kernel whose limit is a Dirac delta function.
In the following, a limiting process for FCCR is examined that has as a limit, the typical conditions for QM with the absence of bound states: both space and time are effectively continuous paramaters, energy is not discretely quantized, a fundamental velocity is bounded, and there exists a constant with the physical dimensions of the gravitational constant, which is bounded and not zero.
The limiting procedure is not unlike that of group contraction, but is distinct in the following way. Group contraction is set up by scaling and therefore parametrizing the group structure constants of a Lie group in such a way that their defining properties are maintained for any value of the parameter(s), then tying the parameter values to representations. The contraction is then the structural limit obtained by passing through a sequence of representations. Here, however, the structural limit is that of a one parameter sequence of Lie groups in their defining representations, where each Lie group in the sequence contains as a subgroup, the preceding group. While this is all stated in terms of Lie groups, the machinery actually used are the associated Lie algebras. In [Section VI], it is shown that Q(n) and P(n) generate the Lie algebra su(n) by commutation. The limits discussed here are only in terms of the fundamental commutation relation (2.8).
Introduce physically dimensioned, nonzero, absolute
and effective constants (functions of n) of:
length: l_0, l(n)
momentum: p_0, p(n)
time: tau_0, tau(n)
action: h-bar, h-bar(n)
with, as usual
h-bar := h/(2 pi)
For FCCR
[Q(n), P(n)] = i G(n)
dress the fundamental operators with multiplicative
scalar functions of the dimension, and of appropriate physical
dimensions:
Q(n) -> ~Q(n) = (l(n)/(sqrt(2))) Q(n)
P(n) -> ~P(n) = (p(n)/(sqrt(2))) P(n) (17.13)
G(n) -> ~G(n) = (h-bar(n))G(n)
and assume that for the fundamental dimensioned constants
l_0 p_0 = h-bar (17.14)
Then FCCR implies for the effective constants that
l(n) p(n) = h-bar(n) (17.15)
If FCCR has as a limit, CCR, when n->infinity
lim h-bar(n) = h-bar (17.16)
lim tau(n) = 0
Without loss of generality let
l(n) := l_0 alpha(n)
p(n) := p_0/beta(n) (17.17)
tau(n) := tau_0 gamma(n)
where alpha(n), beta(n) and gamma(n) are then defined as
dimensionless functions of n.
Then,
(17.18)
h-bar(n) = h-bar (alpha(n)/beta(n))
and
lim (alpha(n)/beta(n)) = 1
lim gamma(n) = 0
If one assumes that a single l(n) transition
requires at least one tau(n) transition, an
effective maximal velocity c(n) can be defined
c(n) := DELTA(~q)(n)/DELTA(~t)(n) (17.20)
= sqrt(pi/(2n)) (l_0 alpha(n))/(tau_0 gamma(n))
where DELTA(~q)(n) is the spectral spacing of ~Q(n).
The gravitational coupling constant G_0 of the Einstein
equations which couples stress-energy-momentum to
geometrical curvature has the physical units (Bridgeman-dimensions):
[L^(3)] [M^(-1)] [T^(-2)] = c^(4) [L] [E^(-1)]
Introduce first, an effective energy quantum
e_0(n) := epsilon(n) h-bar(n)/tau(n) (17.21)
where now epsilon(n) is defined as a dimensionless function of n
which allows the freedom to express the energy scaling and
structure that might be appropriate for some given system
in the QM limit.
Now express an effective gravitational constant as
g_0(n) := c^4(n) (DELTA(~q)(n)/e_0(n)) (17.22)
= c^5(n) tau(n)/e_0(n)
A DIGRESSION ON A TIME OPERATOR & THE EFFECTIVE TIME QUANTUM
The dynamical Ansatz which determines a time operator from
a given Hamiltonian is given in
Theorem 8.20.
Let |E_k> be the eigenbasis of the Hamiltonian H(n).
Construct the unitary cyclic matrix on this eigenbasis, so
C(n) |E_k> = |E_(k-1)>
where the index of eigenvectors is understood mod n.
C(n) is a unitary operator and may therefore be expressed as
C(n) = exp( i omega_H(n) t_H(n) )
where omega_H(n) is a frequency constant attached
to the Hamiltonian, and t_H(n) is the time operator for
the Hamiltonian H(n).
We assume that omega_H(n) is real and that t_H(n) is Hermitean;
this consistent with C(n) being cyclic and so also then unitary.
But we also know from the structure of C(n), that
C(n) = exp( i ((2 pi)/n) N_t)(n) )
where N_t(n) is the number operator in the eigenbasis
of t_H(n). Therefore
((2 pi)/n) N_t(n) = omega_H(n) t_H(n) (17.23)
Then since N_t(n) has an additive spectrum, so does t_H(n) and
omega_H(n) = ((2 pi))/(n tau_0 gamma(n)) (17.24)
Desiderata for n=2 and for the limit of unbounded n:
lim tau(n) = 0
lim Sup Sp( t(n) ) = infinity
implies for large n
O( gamma(n) ) = n^(delta(t))
0 < delta(t) < 1
tau(2) = tau_0
implies
gamma(n) = (n-1)^(-delta(t))
Therefore take
gamma(n) := (n-1)^(-delta(t)) (17.25)
with
0 < delta(t) < 1
Then
Sp( t_H(n) ) -> [0, infinity]
and
omega_H(n) = ((2 pi)/(n tau_0))) (n-1)^(delta(t))
= ((2 pi)/tau_0)) n^((delta(t)-1)), for large n,
and then
lim omega_H(n) = 0 (17.26)
for all values of delta(t).
Recalling that
C(n) = exp( i omega_H(n) t_H(n) )
The limit
lim C(n) = SHA_H(infinity) = SHA
is a one sided shift operator and
is not cyclic; the exponential representation cannot exist in
the limit:
[Section II].
Therefore, one should *not* *even* *expect* that omega_H(n) t_H(n)
will have a proper limit.
The phase in the t(n)-eigenbasis has a limit of a continuous phase
operator with eigenvalues defined on the unit circle in C.
But the t(n)-eigenbasis has no physically acceptable existence in
the limit since UPSILON_H(n) the transformation
connecting the H(n)-eigenbasis with the t(n)-eigenbasis has
no physically acceptable limit.
<n, k|UPSILON_H(n)|n, j> = (1/sqrt(n)) exp( i((2 pi)/n)kj )
END DIGRESSION
Substituting the expression
(17.25)
for gamma(n) into the previous expression
(17.20)
for c(n),
c(n) = (l_0/tau_0) sqrt(pi/2) alpha(n) n^(-1/2) (n-1)^(delta(t)) (17.27)
It is a relativistic desideratum that
lim c(n) = c < infinity
This can be arranged by taking
alpha(n) := n^(1/2) (n-1)^(-delta(t)) (17.28)
In this case
c(n) = c = (l_0/tau_0) sqrt(pi/2) (17.29)
which is independent of n.
Also take
beta(n) = alpha(n) (17.30)
then
h-bar(n) = h-bar (17.31)
which is also independent of n.
Incorporating these results into the effective energy quantum
and the effective gravitational constant:
e_0(n) = (h-bar/tau_0) epsilon(n) (n-1)^(delta(t)) (17.32)
g_0(n) = (c^5 tau_0^2/h-bar) (n-1)^(-2delta(t)) epsilon^(-1)(n) (17.33)
Minimal physical desideratum of the energy quantum:
In the limit, one should have,
lim e_0(n) < infinity
Note that
If we take
epsilon(n) = (n-1)^(-delta(t))
lim (n-1)^(delta(t)) epsilon(n) = 1
otherwise take
epsilon(n) = (n-1)^(-(delta(t)+delta(e)))
where
0 < delta(e)
then
lim (n-1)^(delta(t)) epsilon(n) = 1
Therefore, generally, for a bounded limit, take
epsilon(n) = (n-1)^(-(delta(t)+delta(e))) (17.34)
with
0 < delta(e)
so that
epsilon(n) -> 1, when delta(e) = 0
and
epsilon(n) -> 0, when delta(e) > 0
Now consider the limit of
g_0(n) = (c^5 tau_0^2 /h-bar) (n-1)^(-2 delta(t)) epsilon^(-1)(n)
= (c^5 tau_0^2 /h-bar) (n-1)^((delta(e)-delta(t)))
There are three cases:
Infinite gravitational coupling (physically unacceptable)
i) delta(e) > delta(t), g_0(n) -> infinity
Finite gravitational coupling (independent of n)
ii) delta(e) = delta(t), g_0(n) = (c^5 tau_0^2 /h-bar) (17.35)
Complete gravitational decoupling
iii) delta(e) < delta(t), g_0(n) -> 0
where the only restrictions within these three cases on
the delta(e) and delta(t) are
0 <= delta(e)
(17.36)
0 < delta(t) < 1
Eliminating unphysical case (i), if delta(e) -> 0,
then for case (iii)
e_0(n) -> e_0 not= 0,
g_0(n) -> 0
and for case (ii) (the case of physical choice)
delta(e) = delta(t)
0 < delta(t) < 1
and
e_0(n) -> 0
g_0(n) = (c^5 tau_0^2 / h-bar)
Asymptotic Spectral Radii of operators for large n are:
rho( Q(n) ) = tau (pi/2n) (n-1)/2
rho( P(n) ) = tau (pi/2n) (n-1)/2
rho( G(n) ) = (n-1)
rho( ~Q(n) ) = (l(n)/tau 2)tau(pi/2n) (n-1)/2
rho( ~P(n) ) = (p(n)/tau 2)tau(pi/2n) (n-1)/2
rho( ~G(n) ) = h-bar (n-1)
rho( t_H(n) ) = tau_0 (n-1)^(1-delta(t))
with
0 < delta(t) < 1
For all allowed values of delta(t),
lim rho( ~Q(n) ) = infinity
lim rho( ~P(n) ) = infinity
lim rho( ~t(n) ) = infinity
We have now proved the following
Theorem 17.1:
By Introducing nonzero, absolute and effective
dimensioned constants of:
length: l_0, l(n)
momentum: p_0, p(n)
time: tau_0, tau(n)
action: h-bar, h-bar(n)
with
l_0 p_0 = h-bar
for FCCR
[Q(n), P(n)] = i G(n)
and defining the dressed operators
~Q(n), ~P(n)
as above, so that
[~Q(n), ~P(n)] = i h-bar G(n),
a limit with no bound states, of FCCR can be defined with
FCCR -> CCR
when a "Hamiltonian" H(n) is defined and the associated phase-time
operator t_H(n) is defined by the Ansatz as above
and in
Theorem 8.20
and beyond.
The limits of the operator spectral spacings are:
DELTA(~q(n)) -> 0
DELTA(~p(n)) -> 0
DELTA(t_H)(n) -> 0
DELTA(~q(n)) DELTA(~p(n)) = (pi/(2n)) h-bar
e_0(n) tau(n) = epsilon(n) h-bar
and, therefore that ~Q(n), ~P(n), t_H(n)
all have as a limit operators with unbounded, dense spectra that
are linearly extendible to continuous spectra.
Moreover, in this limit, the constructed constants, c, h-bar, g_0
all have finite nonzero values consistent with their standard
physical interrelationships.
This limiting behavior can be achieved by choosing the undetermined
dimensionless scaling functions to be contrained by:
(17.37)
alpha(n) = alpha(n) = (n-1)^(-delta(t)+1/2)
gamma(n) = (n-1)^(-delta(t))
epsilon(n) = (n-1)^(-2 delta(t))
with
0 < delta(t) < 1
Relation of our assumed physical constants to the Planck units
follows.
The fundamental and Planck units are given by:
Symbol Name Abstract Units cgs approximate value
c Speed of Light [LT^(-1)] 2.998E+10 cm/sec
h-bar Reduced Planck [ET] 1.0546E-27 gm-cm^2/sec
G_0 Gravitation [M^(-1)L^3T^(-2)] 6.673E-8 cm^(3)/gm-sec^(2)
l_P Planck length [L] 1.616E-33 cm
sqrt(h-bar G_0)/c^3)
tau_P Planck time [T] 5.391E-44 sec
sqrt(h-bar G_0)/c^5)
mu_P Planck mass [M] 2.177E-05 gm
sqrt(h-bar c/G_0))
e_P Planck energy [ML^(2)T^(-2)] 1.95669E+16 ergs
sqrt(h-bar c^5/G_0))
p_P Planck momentum [MLT^(-1)] 6.52389E+4 gm-cm/sec
sqrt(h-bar c^3/G_0))
which are related by
e_P tau_P = h-bar
l_P p_P = h-bar
l_P = c tau_P
e_P = mu_P c^2
The standard gravitational constant may then be expressed
G_0 = (tau_P^2 c^5 )/h-bar
From this it appears that we have specific identifications
within the Planck system since our c(n), g_0(n) and h-bar(n) are
actually independent of the dimension n, we identify them with the
fundamental constants upon which the Planck system is founded, then:
tau_0 = tau_P
l_0 = l_P
g_0 = G_0
Our effective quantum of energy, hence also effective mass quantum
has a zero limit, and so quantities of these units are allowed to
be continuously selected in the limit.
Since for these quanta
e_0(n) = m_0(n) c^2
we need only be concerned with the behavior of one of them and
the behavior of the other necessarily follows.
First remember that the exponent delta(t) has *not* yet
been determined, but only constrained by
0 < delta(t) < 1.
and so the other effective constants as functions of n have not
yet been determined. Consider the energy quantum. Given the
relations so far established, write the expression for e_0(n):
e_0(n) = h-bar epsilon(n)/tau_0(n)
= (h-bar/tau_0) (n-1)^(-delta(t))
If for some n finite
e_0(n) = e_P
then
(h-bar/tau_0) (n-1)^(-delta(t)) = sqrt(h-bar c^5/G_0)
Squaring
(h-bar/tau_0^2) (n-1)^(-2 delta(t)) = (c^5/G_0)
(n-1)^(-2 delta(t)) = (c^5/G_0)/(h-bar/tau_0^2)
(n-1)^(2 delta(t)) = (h-bar/tau_0^2)/(c^5/G_0)
(n-1)^(2 delta(t)) = (e_P/tau_0)
(n-1)^(2 delta(t)) = 1
For n > 2, this equation can only hold if the constraint
on delta(t) is violated. Therefore,
e_0(n) not= e_P
for any n > 2. If n = 2, the equation is identically
satisfied and there is no further constraint on delta(t),
we have
e_0(2) = e_P
consequently also then
m_0(2) = e_P/c^2
and
m_0(2) = mu_P
Generally for n > 2
e_0(n) = (h-bar/tau_0) (n-1)^(-delta(t))
m_0(n) = (h-bar/(tau_0 c^2)) (n-1)^(-delta(t))
are energy and mass quanta. But still delta(t) is not determined.
The quanta e_0(n) and m_0(n) converge to zero
as n->infinity more slowly, the smaller delta(t).
Since the entire notion of FCCR started with a variant of the simple harmonic oscillator of QM, it would be rather damning of this mathematical venture should one not address and show explicitly the limiting process which arrives back at the standard quantum mechanical oscillator. Using the above limit as a model for procedure, begin again almost from the beginning. The simple oscillator of QM is a classical system, and so the the maximal velocity c(n) should have an unbounded limit.
Let ....
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