Wigner's proof
Camm Maguire
quantum logic starts with the assertion that only the modulus of the
inner product, |<x,y>|, is measurable. Therefore, unitary *and
anti-unitary* operators are symmetries, the latter of which has real
implications via time-reversal.
The assertion stems from the statement that states x and y are *rays*
in Hilbert space, and invariant with respect to multiplication by an
arbitrary phase.
But is not the real-valued *length* of a "vector" in Hilbert space,
<x,x>=|<x,x>|, also unmeasurable? One might think that from a first
quantization point of view, the length could represent the number of
coherent quanta in the state or some such, but from a second quantized
point of view, this number corresponds to another dimension in the
space, and indeed the state can represent a distribution among various
numbers of quanta. The overall Fock space length, it would appear,
has no meaning and is always just assumed to be one. Indeed, all we
have are probabilities in the end, which must not only be real, but
also bounded by 0 and 1.
If true, then only |<x,y>||<y,x>|/(|<x,x>||<y,y>|) is measurable, and
one can envision symmetry operators defined not only up to an
arbitrary phase, but to an arbitrary scale as well. This probably has
no consequence, but I'd like to ask for corrections to my thinking if
anyone has any.
In principle, this might appear to open up other possible symmetries
to the quantum logic, but I'm not sure if it does. In Wigner's proof,
(an outline of which can be found in Sternberg's book -- "Group Theory
and Physics"), the loose phase-valued function attached to the
operator is found to preserve the real part of the inner product,
leading to the only non-trivial possibility being complex conjugation.
But this logic, it appears, depends on asserting that the symmetry
preserves |<x,y>|, which it would seem is not technically necessary.
One also has a loose scale attached to the symmetry too. My hunch is
that the conclusion of the proof stands, but I have not yet reproduced
all the steps.
Beyond this, most presentations I have read just assert the norm of
all states to be one, yet leave the loose phase as a central extension
to whatever group is being studied, and thereby expanding the
representations available to the projective representations. I'm just
wondering why the scale and phase are treated so asymmetrically. Of
course, one is compact and the other not, which might pose technical
difficulties, especially with finite groups. But still the question
remains, is there any real difference mathematically or physically in
the meaning of the overall phase and length of the wave-function?
Take care,
--
Camm Maguire ca...@maguirefamily.org
==========================================================================
"The earth is but one country, and mankind its citizens." -- Baha'u'llah
xstrangerepx_removex
> But is not the real-valued *length* of a "vector" in Hilbert
> space, <x,x>=|<x,x>|, also unmeasurable? One might think
> that from a first quantization point of view, the length
> could represent the number of coherent quanta in the state
> or some such, but from a second quantized point of view,
> this number corresponds to another dimension in the space,
That sounds not quite right (unless I misunderstood what you meant).
> and indeed the state can represent a distribution among
> various numbers of quanta. The overall Fock space length, it
> would appear, has no meaning and is always just assumed to
> be one.
The important (mathematical) point here is that Fock space is
constructed such that the norms of all vectors therein are
finite (and such that a vector has norm=0 only if it's the
zero vector).
> Indeed, all we have are probabilities in the end,
> which must not only be real, but also bounded by 0 and 1.
>
> If true, then only |<x,y>||<y,x>|/(|<x,x>||<y,y>|) is
> measurable, and one can envision symmetry operators defined
> not only up to an arbitrary phase, but to an arbitrary scale
> as well. This probably has no consequence, but I'd like to
> ask for corrections to my thinking if anyone has any.
Provided that |<x,x>| and |<y,y>| are both finite and nonzero, one can
rescale x and y such that both norms are 1, without affecting the
ratio
you mentioned. Physically, therefore, there's no loss of generality in
taking normalized vectors.
However, if unbounded operators are in play, one can find specific
examples where (say) |<x,y>| or |<x,x>|, etc, are divergent, and yet
the ratio remains finite (by Cauchy-Schwarz). But then one has left
the
familiar realms of linear Hilbert spaces.
I once tried to track through the Wigner proof as given in Weinberg
vol-1, under the relaxed assumption that only the ratio is
well-defined. I soon found that the proof does not go through
(no surprise).
> But still the question remains, is there any real difference
> mathematically or physically in the meaning of the overall
> phase and length of the wave-function?
Consider what unbounded operators do in inf-dim Hilbert spaces... :-)
HTH.
Rock Brentwood
> Greetings! Wigner's famous [unitary or anti-unitary] proof ... The ass=
ertion
> stems from the statement that states x and y are *rays*
> in Hilbert space, and invariant with respect to multiplication by an
> arbitrary phase.
I haven't seen the proof, but your description raises a few point of
interest (to me).
First, since it is cast in the framework of Hilbert space rays, that
means it is actually being posed on a Poisson manifold. The Hilbert
space structure contains a Poisson manifold structure inside the
imaginary part of its bi-linear form, as is fairly well-known.
That means, the assertion and proof may actually having nothing per se
to do with quantum theory or Hilbert space representations at all!
That is, it may be possible to pull it back into the (more general)
non-linear representation theory of Poisson manifolds and symplectic
spaces and cast everything in the language of Poisson manifolds.
Then, it becomes a result that holds both for quantum AND classical
systems.
One way to address the question about the norm is that each sphere of
constant |<x,x>| identifies a separate value for h-bar. (This is the
approach Ashtekar took in his 1990's paper of the Poisson space
geometry of Hilbert spaces); this takes place via the representation |
<x,x>| = h-har.
So, then, it's not so much that |<x,x>| is "unmeasurable", but that it
is simply an invariant. The corresponding gauge degree of freedom, as
Ashtekar explained, is simply the phase. After factoring out the gauge
degree, the result is the reduced Hilbert space -- i.e., the state
space comprising the rays of the original Hilbert space, with the
remaining U(1) phase degree factored out.
But the REAL question being posed here is whether or not the Wigner
result CAN actually be pulled back to the more general framework of
Poisson spaces.
I'm not entirely sure, but I think the answer lies in a direct
examination of the projective extensions of the discrete generators {T
= time reversal, P = parity reversal, I = identity, Z = space-time
reversal = PT}. In the defining representation, this gives you a group
isomorphic to Z2 x Z2 by the table:
I^2 = P^2 = T^2 = Z^2 = I, IP = P = PI, IT = T = TI, IZ =
= Z = ZI,
PT = Z = TP, TZ = P = ZT, ZP = T = PZ.
So ... what are the projective extensions of this group? It turns out
that there are 8 real projective extensions, 2 in the complex domain.
In the projective extension of a group, each product is replaced by a
(real or complex) multiple of the original product; the resulting
assignment of multiples should ensure associativity.
The requirement that everything be associative, means that you
ultimately get 8 possibilities along the following lines:
Ix = x = xI, as before
Z^2 = a, P^2 = b, T^2 = c; with a, b, c = +/- 1
ZP = t'T, PZ = tT, PT = z'Z, TP = zZ, TZ = p'P, ZT = pP
where
zt = z't', pz = p'z', tp = t'p',
z't = b = zt', p'z = c = pz', t'p = d = tp'
comes straight out of the imposition of associativity. This yields the
following solutions
(p', t', z') = s (p, t, z); (b, c, d) = s (zt, pz, pt); where s =
+/- 1.
By redefining P, T, Z with appropriate real multiples, you can always
reduce these parameters to +/- 1. In addition, you can always get one
of the signs (p, t or z) to be +1. So that leaves you with 8
combinations in all.
The 4 with s = -1 extend the algebra {I,P,T,Z} to a QUATERNION
algebra, while those with s = +1 extend the algebra to an Abelian
algebra -- a morphed version of Z2 x Z2, itself.
In the complex domain, you can choose complex multiples of Z, P and T
that reduce all (p,t,z) = (1,1,1); leaving you only with the
quaternion case (s = -1) and Z2 x Z2 case (s = +1).
In the respective cases, T^2 = -1 or T^2 = +1.
The corresponding representation families for Connected-Lorentz X
{I,T} are called, respectively, Sin(+) for T^2 = +1, and Sin(-) for
T^2 = -1 -- where "Sin" means "Spin without the P".
This result is what I was thinking of when seeing your article. I'm
wondering if THIS is what the Wigner result actually has as its basis.
Finally, to expand this to a result for Poisson spaces, we use (more
or less standard) techniques of embedding a Lie algebra into a Poisson
space. This may be thought of as the non-linear extension of the Lie
algebra. Thus, if the generators of a Lie algebra L are {Y_1,
Y_2, ..., Y_n} with Lie brackets [Y_a, Y_b] = f^c_{ab} Y_c (summation
convention used on c), then the Poisson space has coordinates
{Y_1, ..., Y_n} and a Poisson bracket
{f(Y), g(Y)} = f^c_{ab} Y_c (df/dY_a) (dg/dY_b).
In the language of Poisson spaces:
* representations correspond to symplectic spaces
* irreducible representations are symplectic leaves
* matrix representations are generalized to vector fields acting on
the symplectic space
* the generators Y_a are represented by their "Hamiltonian fields
delta_{Y_a} = {_, Y_a}
* central extensions correspond to representations with non-trivial
actions by the Y_a
I'm not sure how projective extensions fit in this scheme, though;
since these are coming out of the discrete part of the Lorentz group,
while the above applies to the Lie algebra and the connected part of
the Lie group.
Igor Khavkine
> Greetings! Wigner's famous proof regarding the symmetries of a
> quantum logic starts with the assertion that only the modulus of the
> inner product, |<x,y>|, is measurable. Therefore, unitary *and
> anti-unitary* operators are symmetries, the latter of which has real
> implications via time-reversal.
The argument is a bit more subtle than that. We do not put the
absolute value bars on all inner products <x,y>. The reason for that
is, although if you write down all possible <x,y>, <x,z>,
<y,z>, ...infinitely many more..., that the *relative* phases of all
these inner products are observable, but any *overall* change in phase
is not. So if the phase of <x,y> is changed while the phase of <x,z>
is kept the same, this kind of transformation has a physically
measurable effect. On the other hand, if the phases of <x,y>, <x,z>,
and all other inner products are changed by the same amount, then this
kind of transformation is unobservable.
> But is not the real-valued *length* of a "vector" in Hilbert space,
> <x,x>=|<x,x>|, also unmeasurable? One might think that from a first
> quantization point of view, the length could represent the number of
> coherent quanta in the state or some such, but from a second quantized
> point of view, this number corresponds to another dimension in the
> space, and indeed the state can represent a distribution among various
> numbers of quanta. The overall Fock space length, it would appear,
> has no meaning and is always just assumed to be one. Indeed, all we
> have are probabilities in the end, which must not only be real, but
> also bounded by 0 and 1.
OK, gotta be really careful here! There is no distinction between the
quantum inner product (the one that yields probability amplitudes) in
first or second or Fock quantization. If you do know a little bit
about first and second quantization, I can see how you might get
confused about that. When wave functions get "promoted" to operators,
what was the wave function inner product gets "promoted" to a so-
called number operator. But after this promotion, the expression
ceases to be an inner product. Feel free to ask followup questions if
this is not completely clear.
> If true, then only |<x,y>||<y,x>|/(|<x,x>||<y,y>|) is measurable, and
> one can envision symmetry operators defined not only up to an
> arbitrary phase, but to an arbitrary scale as well. This probably has
> no consequence, but I'd like to ask for corrections to my thinking if
> anyone has any.
The length squared of a state vector, <x,x>, is associated with the
total probability of all possibilities added up, which is always 100%
or just 1. There are two ways to incorporate this into quantum
mechanics, either you only consider state vectors that have norm 1, or
you declare the length a state vector to be unobservable, which is
exactly as you suggest. The effect is the same. That is, the
physically distinguishable linear operators on the quantum state space
are reduced to the group of unitary operators. In one case you get
unitary operators by rounding up all operators that preserve the inner
product. In the other case, you take all operators that rescale the
length of each vector by the same amount, and then take equivalence
classes of them under a global re-rescaling.
Hope this helps.
Igor
a student
> quantum logic starts with the assertion that only the modulus of the
> inner product, |<x,y>|, is measurable. Therefore, unitary *and
> anti-unitary* operators are symmetries, the latter of which has real
> implications via time-reversal.
>
> The assertion stems from the statement that states x and y are *rays*
> in Hilbert space, and invariant with respect to multiplication by an
> arbitrary phase.
If you want to look at Wigner's theorem from
the point of view of quantum logic, it is
perhaps better to consider it as being about
automorphisms on the Hilbert space which
preserve the "quantum logic" of the space,
i.e., the lattice structure of the closed
subspaces (or, equivalently, the lattice
of projections onto such subspaces). The
theorem says that any such automorphism,
i.e., any 1:1 mapping from the lattice to
itself which preserves orthogonality and
intersections of closed subspaces, can
be realised as a unitary or antiunitary
map.
> But is not the real-valued *length* of a "vector" in Hilbert space,
> <x,x>=|<x,x>|, also unmeasurable? One might think that from a first
> quantization point of view, the length could represent the number of
> coherent quanta in the state or some such, but from a second quantized
> point of view, this number corresponds to another dimension in the
> space, and indeed the state can represent a distribution among various
> numbers of quanta. The overall Fock space length, it would appear,
> has no meaning and is always just assumed to be one. Indeed, all we
> have are probabilities in the end, which must not only be real, but
> also bounded by 0 and 1.
>
> If true, then only |<x,y>||<y,x>|/(|<x,x>||<y,y>|) is measurable, and
> one can envision symmetry operators defined not only up to an
> arbitrary phase, but to an arbitrary scale as well. This probably has
> no consequence, but I'd like to ask for corrections to my thinking if
> anyone has any.
This is a different question, not directly related
to Wigner's theorem.
> In principle, this might appear to open up other possible symmetries
> to the quantum logic, but I'm not sure if it does. In Wigner's proof,
> (an outline of which can be found in Sternberg's book -- "Group Theory
> and Physics"), the loose phase-valued function attached to the
> operator is found to preserve the real part of the inner product,
> leading to the only non-trivial possibility being complex conjugation.
> But this logic, it appears, depends on asserting that the symmetry
> preserves |<x,y>|, which it would seem is not technically necessary.
> One also has a loose scale attached to the symmetry too. My hunch is
> that the conclusion of the proof stands, but I have not yet reproduced
> all the steps.
You are asking if there is a larger symmetry group than the set of
unitary and antiunitary maps, for which the normalised inner product
is invariant. The answer is clearly yes - eg, consider the continuous
map
psi(t) = a(t) psi(0)
for some real-valued function a(t). The
corresponding "Schroedinger" equation is
i hbar d psi/dt = [ihbar d (log a)/dt] psi,
which corresponds to an antiHermitian
"Hamiltonian".
> Beyond this, most presentations I have read just assert the norm of
> all states to be one, yet leave the loose phase as a central extension
> to whatever group is being studied, and thereby expanding the
> representations available to the projective representations. I'm just
> wondering why the scale and phase are treated so asymmetrically. Of
> course, one is compact and the other not, which might pose technical
> difficulties, especially with finite groups. But still the question
> remains, is there any real difference mathematically or physically in
> the meaning of the overall phase and length of the wave-function?
The normalisation of probability requires a
fixed "length" (in the quantum logic picture,
this is part of the content of Gleason's
theorem). The overall phase of a wavefunction
has no physical meaning (but relative phases
are important for superpositions). One way
to see this in standard QM is that the
probabilities and phases are canonically
conjugate variables. This means that
total probability can be conserved only
if the generator of the motion (the
Hamiltonian) is independent of the
overall phase. Since any observable
can be a Hamiltonian, no observable
quantity can be dependent on the
overall phase.