markwh04@yahoo.com
May14-06, 05:00 AM
Originally from: Re: Negative Probability & 19th Century Quantum
Mechanics
http://groups.google.com/group/sci.math/msg/19afb0eb79c9d334?dmode=source
In reference to interpreting negativity of the Wigner function as an
implicit restriction on the allowable subsets of phase space that may
be integrated over:
nobody wrote:
> Now I believe -- but cannot prove -- that for any set A with finite area
> there exists some pure-state Wigner function W such that W(A) < 0. If
> this conjecture is true, it would destroy any hope of deriving the
> uncertainty principle by considering measures defined on familes of sets
> satisfying (1) through (4').
In fact, you're missing the bigger picture entirely: though a Wigner
function need not be positive definite, it is, indeed, closely
associated with a positive definite phase space distribution that goes
a long way toward implementing the general idea and which would have
been right up Boltzmann's alley to arrive at.
Every Wigner function is the inverse Gaussian convolution of a positive
definite density, with widths DP, DQ chosen, such that DP.DQ >=
h-bar/2. This is not quite the sharp phase-space region cut-off idea
originally thought of, but a kind of soft cutoff.
In the process of explaining how this comes about, it will help to
remove some of the cobwebs from the Wigner approach, while also
revealing a clearer picture of where everything is actually coming
from.
Associated with a state W is a linear function W[]. If the state is a
mixed state represented by a density matrix rho, then W[] = Tr(rho []),
and if it's a pure state represented by a Hilbert space vector |psi>,
then W[] = <psi| [] |psi>.
Remarkably, it's gone virtually unnoticed that the Wigner function
W(p,q) for the state has an extremely simple definition: it's simply
the expectation of the delta function,
W(p,q) = W[delta(P-p,Q-q)]
where P and Q are the corresponding operators. This is true in the
sense of weak operator limits and may be resolved by expanding the
delta in integral form and bringing the W[] under the integral.
More remarkable still is that one may proceed to define the operator
form of the Wigner function (or of any observable). For an observable
A(p,q), one defines
A(P,Q) = integral delta(P-p,Q-q)
A(p,q) dp dq.
This effects what is known as "Weyl quantization" or Weyl operator
ordering. Each classical observable A(p,q) being associated with its
corresponding Hermitean operator A(P,Q). For instance, associated with
pq is (PQ+QP)/2 and generally associated with the generating function
exp(ap+bq) is exp(aP+bQ).
One can even write down the action of delta(P-p,Q-q) on a state, as
well as an expansion in terms of an improper basis |x> ... the results
will be quite revealing and show clearly how the peculiar form of the
Wigner formula arises.
The key property of interest here is that
W(P,Q) = h^n rho,
where n is the number of degrees of freedom (where p =
(p_1,p_2,...,p_n), q = (q^1,q^2,...,q^n)). Using this result, one can
then find an expression for the transition probability between two
states,
p(W1,W2) = Tr(rho1 rho2) = W1[rho2] =
W1[W2(P,Q)]/h^n.
Working out the term W1[W2(P,Q)], we get
W1[W2(P,Q)] = integral W1[delta(P-p,Q-q) W2(p,q)] dp dq
= integral W1[delta(P-p,Q-q)] W2(p,q) dp dq
= integral W1(p,q) W2(p,q) dp dq.
Thus, we recover an important theorem:
p(W1,W2) = integral W1(p,q) W2(p,q) dp dq/h^n.
This is where we return to the prior discussion. Had Boltzmann gone on
further, eventually he would have reached the point of reasoning that
every phase space state {p0,q0} is smeared in such a way that there is
an effective widths Dq and Dp in phase space with the average value of
an observable A in the state {q0,p0} being a Gaussian smear
<A>(p0,q0) = integral
A(p,q) exp(-1/2 ((p-p0)/Dp)^2 - 1/2 ((q-q0)/Dq)^2) dp dq/(2
pi Dp Dq).
For the generating function A = exp(ap+bq) this yields
<A>(p,q) = exp(ap+bq+(aDp)^2/2 +
(bDq)^2/2).
Given the smearing, then it follows that if one observes a system to be
in a state {p0,q0} ther eis a non-zero probability that it may actually
be in another state {p,q} nearby. Corresponding to each state {p0,q0}
is a transition probability
p({p0,q0},{p,q}) = exp(-1/2 ((p-p0)/Dp)^2 - 1/2
((q-q0)/Dq)^2) 1/(2 pi Dp Dq).
To implement the Boltzmann quantization of phase space, he assumes that
each degree of freedom has a cell width of size h. This is what
actually represented the first bona fide departure from classical
physics, which admits no such natural quantization. Here, it is
implemented in the integral measure, with
h = 2 pi Dp Dq
and h-bar = h/(2 pi) = Dp Dq.
In fact, it is possible to quantize phase space in this manner and the
corresponding states are already well-known as the "coherent states".
Explicitly, the state {p0,q0} is represented as a coherent state
|p0,q0> which is an eigenvector of the operator (Q/Dq + i P/Dp) with
eigenvalue (q0/Dq + i p0/Dp). And one finds, indeed, that
|<p0,q0|p,q>|^2 = p({p0,q0},{p,q}).
The Wigner function, though it won't be worked out here is a Gaussian
W_{p0 q0}(p,q) = exp(-((p0-p)/Dp)^2 -
((q0-q)/Dq)^2)
up to proportionality. The associated widths are Dp/sqrt(2) and
Dq/sqrt(2) and their product is h-bar/2.
>From this, it follows that for any state W, with density matrix rho,
0 <= <p0,q0|rho|p0,q0> = p({p0,q0},W) =
W_{p0,q0}[W(P,Q)]/h^n.
The last term, as seen above, is the integral
integral W_{p0,q0}(p,q) W(p,q) dp dq/h^n
and inserting the Wigner function for the state {p0,q0}, we get (up to
proportionality)
integral exp(-((p0-p)/Dp)^2 - ((q0-q)/Dq)^2) W(p,q) dp
dq,
which shows that the Gaussian convolution of W with widths Dp/sqrt(2)
Dq/sqrt(2) = h-bar/2 yields a non-negative phase space distribution.
The effect of a Gaussian convolution of a function f(x) to <f>(x), with
a Gaussian of mean mu and width A = sigma^2 can be captured by the
generating function
<exp(kx)> = exp(k (x+mu) + A k^2/2).
This shows that convolutions form a semigroup. Explicitly denoting the
operation by
<[];A,mu>,
one gets
<<[];A,mu>;B,nu> = <[];A+B,mu+nu>.
>From this, it follows that further convolution of the smeared Wigner
function above will preserve the non-negativity, and that ANY
convolution of widths Dp, Dq such that Dp.Dq >= h-bar/2 will yield a
non-negative distribution.
Formally, one may deefine the inverse of <[];A,mu> by <[],-A,-mu>,
though this need not be well-defined except over a smaller space of
functions. The corresponding generating function is <exp(kx);-A,-mu> =
exp(k(x-mu) - A k^2/2> and it's here you can see where the negativity
of the Wigner function arises. For we have,
<x^2;-A,-mu> = (x+mu)^2 - A,
which is negative if x+mu is sufficiently small.
The Wigner state is then the inverse convolution of its smeared variant
with the width Dp and Dq given above.
In this way, it is seen that a quantum state may be equivalently
represented by a Wigner function or by a non-negative phase space
probability distribution. Both yield equivalent information, each form
being convertible into the other.
Mechanics
http://groups.google.com/group/sci.math/msg/19afb0eb79c9d334?dmode=source
In reference to interpreting negativity of the Wigner function as an
implicit restriction on the allowable subsets of phase space that may
be integrated over:
nobody wrote:
> Now I believe -- but cannot prove -- that for any set A with finite area
> there exists some pure-state Wigner function W such that W(A) < 0. If
> this conjecture is true, it would destroy any hope of deriving the
> uncertainty principle by considering measures defined on familes of sets
> satisfying (1) through (4').
In fact, you're missing the bigger picture entirely: though a Wigner
function need not be positive definite, it is, indeed, closely
associated with a positive definite phase space distribution that goes
a long way toward implementing the general idea and which would have
been right up Boltzmann's alley to arrive at.
Every Wigner function is the inverse Gaussian convolution of a positive
definite density, with widths DP, DQ chosen, such that DP.DQ >=
h-bar/2. This is not quite the sharp phase-space region cut-off idea
originally thought of, but a kind of soft cutoff.
In the process of explaining how this comes about, it will help to
remove some of the cobwebs from the Wigner approach, while also
revealing a clearer picture of where everything is actually coming
from.
Associated with a state W is a linear function W[]. If the state is a
mixed state represented by a density matrix rho, then W[] = Tr(rho []),
and if it's a pure state represented by a Hilbert space vector |psi>,
then W[] = <psi| [] |psi>.
Remarkably, it's gone virtually unnoticed that the Wigner function
W(p,q) for the state has an extremely simple definition: it's simply
the expectation of the delta function,
W(p,q) = W[delta(P-p,Q-q)]
where P and Q are the corresponding operators. This is true in the
sense of weak operator limits and may be resolved by expanding the
delta in integral form and bringing the W[] under the integral.
More remarkable still is that one may proceed to define the operator
form of the Wigner function (or of any observable). For an observable
A(p,q), one defines
A(P,Q) = integral delta(P-p,Q-q)
A(p,q) dp dq.
This effects what is known as "Weyl quantization" or Weyl operator
ordering. Each classical observable A(p,q) being associated with its
corresponding Hermitean operator A(P,Q). For instance, associated with
pq is (PQ+QP)/2 and generally associated with the generating function
exp(ap+bq) is exp(aP+bQ).
One can even write down the action of delta(P-p,Q-q) on a state, as
well as an expansion in terms of an improper basis |x> ... the results
will be quite revealing and show clearly how the peculiar form of the
Wigner formula arises.
The key property of interest here is that
W(P,Q) = h^n rho,
where n is the number of degrees of freedom (where p =
(p_1,p_2,...,p_n), q = (q^1,q^2,...,q^n)). Using this result, one can
then find an expression for the transition probability between two
states,
p(W1,W2) = Tr(rho1 rho2) = W1[rho2] =
W1[W2(P,Q)]/h^n.
Working out the term W1[W2(P,Q)], we get
W1[W2(P,Q)] = integral W1[delta(P-p,Q-q) W2(p,q)] dp dq
= integral W1[delta(P-p,Q-q)] W2(p,q) dp dq
= integral W1(p,q) W2(p,q) dp dq.
Thus, we recover an important theorem:
p(W1,W2) = integral W1(p,q) W2(p,q) dp dq/h^n.
This is where we return to the prior discussion. Had Boltzmann gone on
further, eventually he would have reached the point of reasoning that
every phase space state {p0,q0} is smeared in such a way that there is
an effective widths Dq and Dp in phase space with the average value of
an observable A in the state {q0,p0} being a Gaussian smear
<A>(p0,q0) = integral
A(p,q) exp(-1/2 ((p-p0)/Dp)^2 - 1/2 ((q-q0)/Dq)^2) dp dq/(2
pi Dp Dq).
For the generating function A = exp(ap+bq) this yields
<A>(p,q) = exp(ap+bq+(aDp)^2/2 +
(bDq)^2/2).
Given the smearing, then it follows that if one observes a system to be
in a state {p0,q0} ther eis a non-zero probability that it may actually
be in another state {p,q} nearby. Corresponding to each state {p0,q0}
is a transition probability
p({p0,q0},{p,q}) = exp(-1/2 ((p-p0)/Dp)^2 - 1/2
((q-q0)/Dq)^2) 1/(2 pi Dp Dq).
To implement the Boltzmann quantization of phase space, he assumes that
each degree of freedom has a cell width of size h. This is what
actually represented the first bona fide departure from classical
physics, which admits no such natural quantization. Here, it is
implemented in the integral measure, with
h = 2 pi Dp Dq
and h-bar = h/(2 pi) = Dp Dq.
In fact, it is possible to quantize phase space in this manner and the
corresponding states are already well-known as the "coherent states".
Explicitly, the state {p0,q0} is represented as a coherent state
|p0,q0> which is an eigenvector of the operator (Q/Dq + i P/Dp) with
eigenvalue (q0/Dq + i p0/Dp). And one finds, indeed, that
|<p0,q0|p,q>|^2 = p({p0,q0},{p,q}).
The Wigner function, though it won't be worked out here is a Gaussian
W_{p0 q0}(p,q) = exp(-((p0-p)/Dp)^2 -
((q0-q)/Dq)^2)
up to proportionality. The associated widths are Dp/sqrt(2) and
Dq/sqrt(2) and their product is h-bar/2.
>From this, it follows that for any state W, with density matrix rho,
0 <= <p0,q0|rho|p0,q0> = p({p0,q0},W) =
W_{p0,q0}[W(P,Q)]/h^n.
The last term, as seen above, is the integral
integral W_{p0,q0}(p,q) W(p,q) dp dq/h^n
and inserting the Wigner function for the state {p0,q0}, we get (up to
proportionality)
integral exp(-((p0-p)/Dp)^2 - ((q0-q)/Dq)^2) W(p,q) dp
dq,
which shows that the Gaussian convolution of W with widths Dp/sqrt(2)
Dq/sqrt(2) = h-bar/2 yields a non-negative phase space distribution.
The effect of a Gaussian convolution of a function f(x) to <f>(x), with
a Gaussian of mean mu and width A = sigma^2 can be captured by the
generating function
<exp(kx)> = exp(k (x+mu) + A k^2/2).
This shows that convolutions form a semigroup. Explicitly denoting the
operation by
<[];A,mu>,
one gets
<<[];A,mu>;B,nu> = <[];A+B,mu+nu>.
>From this, it follows that further convolution of the smeared Wigner
function above will preserve the non-negativity, and that ANY
convolution of widths Dp, Dq such that Dp.Dq >= h-bar/2 will yield a
non-negative distribution.
Formally, one may deefine the inverse of <[];A,mu> by <[],-A,-mu>,
though this need not be well-defined except over a smaller space of
functions. The corresponding generating function is <exp(kx);-A,-mu> =
exp(k(x-mu) - A k^2/2> and it's here you can see where the negativity
of the Wigner function arises. For we have,
<x^2;-A,-mu> = (x+mu)^2 - A,
which is negative if x+mu is sufficiently small.
The Wigner state is then the inverse convolution of its smeared variant
with the width Dp and Dq given above.
In this way, it is seen that a quantum state may be equivalently
represented by a Wigner function or by a non-negative phase space
probability distribution. Both yield equivalent information, each form
being convertible into the other.