First we rework some of the standard details with the explicit assumption that we are dealing with finite matrices.
Let x, y and z be Hermitean matrices defined as operators on the n-dimensional complex Hilbert space Hilb(n) equipped with the usual Euclidean inner product with respect to which x, y and z are Hermitean, such that:
[x, y] = i z (13.1)
where the commutator is defined using an associative product
[x, y] := xy - yx
Then all spectral values of x, y, and z are real. Further assume that x and y have non-degenerate spectra, so that there exists for x and for y, an orthonormal basis of eigenvectors spanning Hilb(n). Define for an arbitrary vector |psi>, the expectation value, rms uncertainty and translated operator for x:
<psi|x|psi>
<x> := ----------- (13.2a)
<psi|psi>
(DELTA x)^2 := < ( x - <x> )^2 > = < x^2 > - <x>^2 (13.2b)
X := x - <x> (13.2c)
so that
(DELTA x)^2 = < X^2 >
(DELTA y)^2 = < Y^2 >
and
[X, Y] = iz
Note that no normalization and consequent
passing to the associated projective ray space of
Hilb(n) has been explicitly assumed; yet, expectation values
are, as in QM invariant under an overall phase transformation of the vector.
Then, the Schwartz inequality
|<x|x>|^2 |<y|y>|^2 >= |<x|y>|^2 (13.3)
implies
<X^2> <Y^2> >= |<XY>|^2 (13.4)
We have identically
XY = (1/2)[X, Y] + (1/2){X, Y}
where the Jordan product or anticommutator is defined as,
{X, Y} := XY + YX
Then taking the expectation value and using (13.1),
<XY> = (1/2)<iz> + (1/2)<{X, Y}>
we have from (13.4)
(DELTA x)^2 (DELTA y)^2 >= |(i/2)<z> + (1/2)<{X, Y}>|^2 (13.5)
where
(1/2)<{X, Y}> = (1/2)<{x, y}> - <x><y> (13.6)
Equation
(13.6)
is frequently said to be the
quantum analog of the covariance of x and y.
That the analogy is not perfect is the conclusion of
[Margenau 1963].
Margenau states that in QM, if
(13.6)
is adopted as a probablistic
covariance, then,
with {x_i} and {y_j} being the associated spectral sets,
joint probabilities
p(x_iy_j; psi)
calculated in the standard relation
Cov(X, Y) = SIGMA p(x_i y_j; psi) x_i y_j - <x><y>
i j
can be negative.
Here, since X and Y are both Hermitean, it is easy to show that {X, Y}
is Hermitean. Its expectation values will then be real. So the left
hand side of
(13.5)
is the absolute value of a complex number in Cartesian
form. Equation
(13.5)
then becomes
(DELTA x)^2 (DELTA y)^2 >= (1/4)<z>^2 + (1/4)<{X, Y}>^2 (13.7)
So, in particular for any n, we have, particularizing to Q(n) and P(n),
(DELTA Q(n))^2 (DELTA P(n))^2 >=
1/4 <G(n)>^2 + 1/4 <{Q(n), P(n)}>^2 (13.8a)
From Corollary 8.2.2 in [Section VIII], we see the existence of |psi> for which <psi|psi> not= 0,
and <psi|G(n)|psi> not= 0, but for which <psi|{X, Y}|psi> = 0.
Note:
Formalistically, if the vector in question |psi> is an eigenvector of either x or y, then <{X, Y}> = 0. [lemma 8.1]
Further then, <G> = 0. The same phenomenon appears in the usual uncertainty derivation for QM. [Davidson 1965]. In QM, say x|xi> = xi |xi>, then from the canonical commutation relation [x, y] = I, one proves 0 = 1. The error there is a domain problem, assuming that x is selfadjoint with respect to y|xi>. [Carruthers 1968] Discussion of the situation with FCCR is of a different nature.
If <G> = 0, it appears that the uncertainty relation is violated. Prototypical vectors for which this holds are the eigenvectors of Q(n) and P(n). Assume that |n, k> are physically realizable. Then the diagonalizing transformation for Q(n) is real orthogonal. If G(n) is to be form invariant under some group of canonical transformations [Section XI], the diagonalizing [lemma 8.1] transformation for Q(n) is not allowed; similarly the diagonalizing transformation for P(n) being unitarily equivalent to the diagonalizing transformation for Q(n), is also is not allowed. Physically then, although the eigenvectors of Q(n) and P(n) exist mathematically, they are not physically accessible by a coordinate transformation from vectors for which <G> > 0 or for <G> < 0, this in analogy to the unreachability of the light cone from spacelike or timelike points of a Minkowskian space under Lorentz transformations. This introduces a notion of intrinsic dispersion attached to the vectors of "position" and "momentum" when viewed from a timelike or spacelike point of view. On the lightcone, however, the notions of position and momentum become intrinsically discrete. In [Zeeman 1967] it is argued that the physically correct topology on the lightcone should indeed be discrete. The question then arises as to how close one can come to the eigenvectors without destroying the form invariance of G(n). The uncertainty relation may be saved from violation by some mechanism, but the specifics of the rescue are not yet obvious. Notice that it is not saved by the "covariance" term since
<q(n, k)|{Q(n), P(n)}|q(n, k)>
= 2 q(n, k) <q(n, k)|P(n)|q(n, k)>
= 0
With this, the foregoing caveat, (13.8a) becomes
(DELTA Q(n))^2 (DELTA P(n))^2 >= (1/4) <G(n)>^2 (13.8b)
Which expresses the uncertainty relation for FCCR
in analogy to QM.
There is, however, one major difference from QM.
G(n) is not the n-dimensional identity, and is not
a strictly positive nor a strictly negative operator.
Therefore, <psi|G(n)|psi>, and consequently,
<G(n)> := <psi|G(n)|psi>/<psi|psi>
can be positive, negative or zero.
In fact given the known spectrum of G(n), it is clear that
-(n-1) <= <G(n)> <= +1
so the spectral radius of G(n)
rho( G(n) ) = (n-1)
Simply because the proof of the uncertainty relation fails does not, of course, mean that the concept of uncertainty has failed. Because the product of two uncertainties is zero does not mean that both are zero. It happens that in FCCR context one uncertainty may be zero but the other turns out to be of a magnitude close to 15% of the full spectral range of its associated operator. See Corollary 8.10.1. It may be appropriate to redefine the statement of the uncertainty principle to show that another combination of the uncertainties of complementary pairs is always greater than some "constant". It appears, for instance, that for any of the states where the usual statement fails and where
[X, Y] not= 0 we will have ( (DELTAX) + (DELTAY) )^2 > 0since the quantity on the left can only vanish if both uncertainties vanish. The uncertainties of the primary operators on specific eigenvectors can be easily calculated taking into account that |n, k>, |q(n, k)> and |p(n, k)> are orthonormal bases. The uncertainty in some quantity Z(n) on the normalized vector |psi> is then defined by
DELTA(Z(n): |psi>) := <psi| Z^2(n) |psi> - <psi| Z(n) |psi>^2
Then,
DELTA(G(n): |q(n, k)>) = sqrt(n-1)
DELTA(G(n): |p(n, k)>) = sqrt(n-1)
DELTA(Q(n): |q(n, k)>) = 0
DELTA(P(n): |p(n, k)>) = 0
DELTA(G(n): |n, k>) = 0
DELTA(N(n): |n, k>) = 0
since
<n, k| Q(n) |n, k> = 0
<n, k| P(n) |n, k> = 0
<n, k| Q^2(n) |n, k> = (1/2)<n, k| H_osc(n) |n, k>
<n, k| P^2(n) |n, k> = (1/2)<n, k| H_osc(n) |n, k>
H_osc(n) = N(n) + (1/2)G(n)
= (1/2)(Q^2(n) + P^2(n))
Also then,
DELTA^2(Q(n): |n, k>) = (1/2)<n, k| H_osc(n) |n, k>
DELTA^2(P(n): |n, k>) = (1/2)<n, k| H_osc(n) |n, k>
From Theorem 8.3
DELTA^2(Q(n): |p(n, k)>) =
DELTA^2(P(n): |q(n, k)>) =
<q(n, k)|P^2(n)|q(n, k)> =
1
= SIGMA -----------------------
j not= k ( q(n, k) - q(n, j) )^2
which for large n, by the asymptotic equal spacing rule of
Theorem 9.4,
1
= (4n/ pi^2) SIGMA -----------
j not= k ( k - j )^2
By a change of summation index l = k - j,
1
SIGMA ----------- =
j not= k ( k - j )^2
k (n-1-k)
= ( SIGMA + SIGMA ) l^(-2)
l=1 l=1
By [Jolley 1961] #336
infinity
SIGMA j^(-2) = ((pi^2)/6)
j=0
and [Jolley 1961] #356
n
SIGMA j^(-2) =
j=0
infinity
((pi^2)/6) - SIGMA [l^2 B(n+l l)]^(-1)
l=0
where B(. .) is a binomial coefficient, we have
k (n-1-k)
( SIGMA + SIGMA ) l^(-2)
l=1 l=1
infinity
= ((pi^2)/3) - SIGMA [l^2 B(n+l l)]^(-1) + [l^2 B(n-1-k+l l)]^(-1)
l=0
For very large n, we can approximate the uncertainty independently
of k by
DELTA^2(P(n): |q(n, k)>) = (4n/(pi^2))((pi^2)/3)
or
DELTA(P(n): |q(n, k)>) = sqrt(4n/3)
The uncertainty, therefore has an asymptotic growth with sqrt(n), that is
of the same order as asymptotic growth of the spectral radius.
Given the boundedness of the spectrum, this signals a maximum uncertainty.
The product uncertainty of P(n) and Q(n)
in one of the |n, k>
DELTA(Q(n): |n, k>) DELTA(P(n): |n, k>)
= (1/2)<n, k| H_osc(n) |n, k>
| (1/2)(k + 1/2), k not= n-1
= |
| (1/2)(n - 1), k = n-1
As a comparison, for the QM harmonic oscillator:
DELTA(Q: |k>) DELTA(P: |k>) = E_k/omega = h-bar (k + (1/2))
An Alternative to Failure in the Proof of Uncertainty Relation
Consider the simplest case n=2. Use "dressed" operators
Q#(2) = a l_0 Q(2)
P#(2) = (1/a) p_0 P(2)
where
p_0 l_0 = h-bar
so
[Q#(2), P#(2)] = i h-bar G(2)
(Note similarity with the DeBroglie relation between wave length
and the momentum of a photon.)
and p_0 has physical dimensions of momentum while l_0 has physical
dimensions of length, and a is an arbitrary real constant.
If we pick one of the eigenstates of Q#(2), and compute the
uncertainty of P#(2) in it,
DELTA(P#(2): |q>)^2 = h-bar^2 (2/l_0^2)
while for the spacing between the two available eigenvalues
DELTA(q#(2))^2 = (sqrt(2) l_0)
Therefore
DELTA(P#(2): |q>) DELTA(q#(2))^2 = 2 h-bar
reinstating a concept of an uncertainty relation when an
eigenstate of Q#(2) is chosen.
Following this a bit further, for a traversal between q(2, 0) and
q(2, 1) classically, the average momentum of a massive
particle is
Av( p ) = m (delta t) sqrt(2) l_0
If the uncertainty in the momentum allows the 2 q-points
to be reachable from each other as a virtual transition,
the radius of the momentum is greater than or equal to
the average momentum;
this is consistant with assuming that the spacing (DELTA q#(n))
can be treated as an uncertainty.
DELTA(P#(2))^2 >= Av( p )
From this, recognizing then that (delta t) = tau_0
(h-bar tau_0)/(2l_0^2) >= m
placing an upper bound on the mass. If we take c = l_0/tau_0, then
h-bar /(2cl_0) >= m
is also expressed as an upper bound on the rest energy
(1/2)(h-bar c)/(2l_0) >= (m c^2)
Considering the alternative of using an uncertainty in position
that is defined by the spacing together with the momentum,
uncertainty calculated in the standard way for large values of n.
As an example, if n is odd and the eigenstate of Q(n) is the central
one associated with eigenvalue 0, the calculated uncertainty is
approximated by
DELTA(P#(n): |q=0>) approx=
( pi h-bar /sqrt(3) ) ( DELTA(q#(n)) )^(-1)
and
DELTA(P#(n): |q=0>) (DELTA(q#(n))) approx=
( pi h-bar /sqrt(3) )
so that for any value of n, an uncertainty relation exists even
in the eigenstate cases that fail absolutely in QM.
If as in the case n=2, for and traversal from q_k to either q_k+1 or q_k-1, classically considered, the average momentum can be expressed
<p#> = m (DELTA(q#(n)))/(delta t)
where m is an effective mass and (delta t) is the time quantum
at arbitrary n.
Again, if
DELTA(P#(n): |q=0>) approx= <p#>
then there are upper bounds on mass and energy:
(h-bar (delta t)/(sqrt(6)) (DELTA(q#(n)))^(2) approx= m
or
(h-bar c^2 (delta t)/(sqrt(6)) (DELTA(q#(n)))^(2) approx= mc^2
To Normalize or not to Normalize
The usual thing to do in QM with Copenhagen interpretational addendum, is to say, that we will normalize the vectors of the Hilbert space, that is adopt the projective space of vectors, say that physical equivalence obtains under unitary transformations, which preserves the Hermiticty of the operators associated with observables, and then note that preservation of the normalization under unitary transformations is an expression of the conservation of probability.
If we try to adopt a normalization condition, say
<psi| G(n) |psi> = 1
or
<psi|psi> = 1
for all |psi>, the first
normalization condition
will not be preserved by the group U(n) of
unitary transformations, since
G(n) is not generally form invariant under U(n).
It is preserved by another group that is addressed in
[Section XI].
The relevant comment here, however, is that
if it is required that physical invariance be associated
with the invariance of G(n), and therefore a form invariance
of FCCR, and if the |n, k> are assumed to be
physically realizable, then the diagonalizing transformation
XI(n) of Q(n), nor that of P(n) are elements
of the invariance group, and so the eigenvectors |q(n, k)>
and |p(n, k)> are not physically realizable.
This I see as more of a fortuity than a problem,
given the aforementioned relativistic interpretation.
The second normalization condition will be preserved by U(n) in a defining representation on Hilb(n), but neither the uncertainty bound, nor the form invariance of FCCR will be preserved. Blindly applying the standard QM normalization condition seems not the route to take.
Consider the influence on equation
(13.8b)
by the restriction to a subspace,
enforced in the |n, k> basis.
We break the calculation of
(13.5)
according to the "positive subspace" spanned by
the N(n) eigenbasis
|n, 0>, |n, 1>, ... |n, n-2>, where <n, k| G(n) |n, k> = 1 > 0
Particularizing the calculation of the uncertainty
in [Appendix G] to FCCR generators Q(n) and P(n):
| Q(n-1) sqrt(n-1) |n-1, n-2> |
| |
Q(n) = | |
| |
| sqrt(n-1) <n-1, n-2| 0 |
| P(n-1) -i sqrt(n-1) |n-1, n-2> |
| |
P(n) = | |
| |
| sqrt(n-1) <n-1, n-2| 0 |
Then
<Q^2(n)> = <x|Q^2(n-1)|x>
+ (n-1) |<x|n-1, n-2>|^2
+ 2 sqrt(n-1) Re( w<x|Q(n-1)|n-1, n-2> )
+ (n-1) |w|^2
<P^2(n)> = <x|P^2(n-1)|x>
+ (n-1) |<x|n-1, n-2>|^2
+ 2 sqrt(n-1) Im( w<x|P(n-1)|n-1, n-2> )
+ (n-1) |w|^2
and
<Q(n)>^2 = <x|Q(n-1)|x>^2
+ 4 sqrt(n-1) <x|Q(n-1)|x> Re( w <x|n-1, n-2> )
+ 4 (n-1) [Re( w <x|n-1, n-2> )]^2
<P(n)>^2 = <x|P(n-1)|x>^2
+ 4 sqrt(n-1) <x|P(n-1)|x> Im( w <x|n-1, n-2> )
+ 4 (n-1) [Re( w <x|n-1, n-2> )]^2
Uncertainty (DELTA A) is defined by:
(DELTA A)^2 = <A^2> - <A>^2
Most generally then:
(DELTA Q(n))^2 = <x|Q^2(n-1)|x> - <x|Q(n-1)|x>^2
+ (n-1) |<x|n-1, n-2>|^2
+ 2 sqrt(n-1) Im(w<x|Q(n-1)|n-1, n-2>)
+ (n-1) |w|^2
- 4 sqrt(n-1) <x|Q(n-1)|x> Re( w <x|n-1, n-2> )
- 4 (n-1) [Re( w <x|n-1, n-2> )]^2
and
(DELTA P(n))^2 = <x|P^2(n-1)|x> - <x|P(n-1)|x>^2
+ (n-1) |<x|n-1, n-2>|^2
+ 2 sqrt(n-1) Im( w<x|P(n-1)|n-1, n-2> )
+ (n-1) |w|^2
- 4 sqrt(n-1) <x|P(n-1)|x> Im( w <x|n-1, n-2> )
- 4 (n-1) [Re( w <x|n-1, n-2> )]^2
But, from the general expression with w = 0;
(DELTA A)^2 = <x|M^2|x> - <x|M|x>^2 + <x|v><u|x>
If A is also Hermitean,
(DELTA A)^2 = <x|M^2|x> - <x|M|x>^2 + <x|v><v|x>
Again particularizing to Q(n) and P(n),
(DELTA Q(n))^2 = <x|Q^2(n-1)|x>
- <x|Q(n-1)|x>^2
+ (n-1)|<x|n-1, n-2>|^2
(DELTA P(n))^2 = <x|P^2(n-1)|x>
- <x|P(n-1)|x>^2
+ (n-1)|<x|n-1, n-2>|^2
Ergo, if we consider computing the uncertainty of either Q(n) or P(n) on some vector that is restricted to the n-1 dim "positive subspace", i.e., with w = 0, there is still a contribution to the uncertainty from the |v> and <u| of Q(n) or P(n) unless the last component of |x> is also zero. If this last component of |x> does not vanish, the contribution to the uncertainty is always positive.
Note that |v> <u| and |x> are contained in the (n-1)-dimensional subspace. For the Hermitean Q(n) and P(n), the |v> for all n > 2, have as the only nonzero entry their "last element". The problem is that the G(n)-positive subspace is invariant under neither the action of Q(n) nor the action of P(n).
Let A(m) be a product of the form Q^a(n)P^b(n) or P^b(n)Q^a(n) where a+b = m, then if A(m)|psi> is to be in the positive subspace, |psi> must be in the n-m dimensional subspace spanned by |n, 0>, ..., |n, n-1-m>. Thus, for no |psi> can A(n-1)|psi> be in the positive subspace.
The moral: to make a "normal" uncertainty (a quadratic operator) that escapes the indefiniteness of the quadratic form G(n), by restricting the state vector, the restriction must be that the last TWO components of the vector in |n,k>-representation, must vanish.
Notice that these two components are exactly those that are acted upon by a noncompact subgroup "the boosts" of U(n-1, 1) when it is taken as kinematical invariance group (13.5) [Section XI]. Also, the su(2) algebra formed by X_1(n), X_2(n) and X_3(n) acting on the 2-dimensional subspace "at the bottom" acting on this subspace. [Section IV].
Email me, Bill Hammel at: