... do not in fact have anything to do with each other
While writing my eLetter regarding Alex Hankey's (1) support and defence of Lionel Milgrom (2), I took a look at a short letter written by Hankey entitled “Weak Quantum Theory: Satisfied by Quantized Critical Point Fluctuations” (3). Only the first page is freely available, but I’m assuming his reference to Walach is Ref. (4) and the reference to weak quantum theory is Atmanspacher et al. (5).
Atmanspacher et al. (5) give the following as their Eq. (1),
in which ℏ = h/2π (and h is Planck's constant). This is the commutator for two complementary observables P and Q. An example of two such observables would be the position in the x-direction, x, and the momentum in the same direction, px:
[x,px] = xpx − px x = iℏ
The momentum operator px actually has the form
and is intended to operate on a wavefunction ψ like this:
in which ∂ψ/∂x just means “the rate of change of the wavefunction in the x direction”.
The position operator x is in fact just x, so we can expand Eq. 2 like this:
The second term now contains “the rate of change of ‘x times the wavefunction’ in the x direction” and the general rule for dealing with this is
where f and g are any general functions of x, so
[x,px]ψ = −iℏx|| + iℏx|| + iℏψ|| (7)|
in which the first two terms cancel out and ∂x/∂x = 1 so we are left with
However, Hankey actually writes that
“In weak quantum theory, observables or weak quantum fields are postulated not to obey the usual quantum commutation relations, characteristic of ordinary quantum fields:
Firstly, we note that this equation appears to be referring to wavefunctions rather than observables, and that the distinction seems to be lost on Hankey. (Operators are functions which operate on wavefunctions to produce other wavefunctions, they are not in themselves wavefunctions.) Secondly, he has h on the right hand side instead of ℏ so he is out by a factor of 2π. Thirdly, he is taking the commutator of a wavefunction ψ and its Hermitian adjoint ψ† (or possibly the complex conjugate ψ* is intended as in Eq. (4)) rather than the commutator of two complementary observables.
Now ψ* ψ = |ψ|2 and for a normalized wavefunction,
For concreteness let us consider the particle in a 1-dimensional box (length L) which has wavefunctions of the form
ψn = || sin||⎛|
where n is a positive integer. In this case, ψn* = ψn so it is easy to see that [ψn , ψn*]=0 and Hankey's Eq. (1) is invalidated. For completeness, using
(i.e. changing the sign of the imaginary part of px which is all of it) we find
[px,px*]ψ = px px* ψ − px* px ψ = −iℏ×iℏ || ψ
+iℏ×iℏ || ψ = 0
since i and ℏ are constants (i.e. they do not change with x). An operator A is actually “self-adjoint” if A† = A. Operators corresponding to observable quantities always have this property, and would always give [A,A†]=0. Ladder operators, which raise or lower the eigenvalues of other operators, are not self-adjoint. In fact, if X is a lowering operator for N then X† is a raising operator for N (and vice-versa). But in this case it should be generally the case that X and X† commute. Assuming [N,X]=cX (c is real and positive) and N|n〉=n|n〉,
NX|n〉||=|| (XN + [N,X])|n〉 || (14)|
| ||=|| (XN + cX)|n〉 || (15)|
| ||=|| XN|n〉 + cX|n〉 || (16)|
| ||=|| Xn|n〉 + cX|n〉 || (17)|
| ||=|| (n+c)X|n〉
so if N operates on the state |n〉 to give the eigenvalue n, then X acts on |n〉 to give a state X|n〉 on which N operates to give n+c, where c is the commutator of N and X. X† acts to lower the eigenvalue by c, so [N,X†]=−cX†:
NX†|n〉||=|| (X†N + [N,X†])|n〉 || (19)|
| ||=|| (X†N − cX†)|n〉 || (20)|
| ||=|| X†N|n〉 − cX†|n〉 || (21)|
| ||=|| X†n|n〉 − cX†|n〉 || (22)|
| ||=|| (n−c)X†|n〉
We find that the eigenvalue of the N operator on a state |n〉 which has been raised and then lowered,
N X† X |n〉||=|| (−cX†+X†N)X|n〉 || (24)|
| ||=|| −cX†X|n〉+X†NX|n〉 || (25)|
| ||=|| −cX†X|n〉+X†(n+c)X|n〉 (using Eq. (18)) || (26)|
| ||=|| −cX†X|n〉+(n+c)X†X|n〉 || (27)|
| ||=|| nX†X|n〉
is n, as it was originally. We can also lower and then raise the state (as long as we are not starting in the lowest possible state),
N X X† |n〉||=|| (cX+XN)X†|n〉 || (29)|
| ||=|| cXX†|n〉+XNX†|n〉 || (30)|
| ||=|| cXX†|n〉+X(n−c)X†|n〉 (using Eq. (23)) || (31)|
| ||=|| cXX†|n〉+(n−c)XX†|n〉 || (32)|
| ||=|| nXX†|n〉
and again we obtain n, meaning that it does not matter what order we apply the raising and lowering operators, which means that they commute: [X,X†]=XX†−X†X=0. So, we have disproved Hankey's Eq. (1) for particular cases of self-adjoint and non-self-adjoint operators. The commutator, Eq. (1), only applies to pairs of complementary operators such as position and momentum. It does not apply to an operator and its adjoint, or to a wavefunction and its complex conjugate, at least in the cases I’ve just examined.
Hankey's Eq. (3) looks a bit like the standard deviation (ΔX = √〈X2〉−〈X〉2 for some operator X, where 〈X〉=〈ψ|Xψ〉) and as he correctly points out all the quantities in his Eq. (3) are numbers and so “commute with everything”. They are all actually real numbers, equal to their complex conjugates, so it therefore makes no sense whatsoever for him to try to construct the commutator in Eq. (4) or to substitute in from Eq. (3). In fact he seems to have lost the distinction between wavefunctions and operators again.
This was all an attempt to find an object which obeys a “more general commutation relation” which does not involve Planck's constant (but presumably a much larger number) but Hankey's Eq. (5) contains h anyway (6), so it's not as if he's completely escaped from the constraints applied by Planck's constant, despite have escaped from the constraints of reality.
- A. Hankey, Evid. Based Comp. Alt. Med. (eCAM) Advance Access published online on May 14, 2008.
- L. R. Milgrom, Evid. Based Comp. Alt. Med. (eCAM) 4, 7 (2007).
- A. Hankey, J. Alt. Comp. Med. 12, 105 (2006).
- H. Walach, Forsch. Komplementmed. 10, 192 (2003).
- H. Atmanspacher, H. Römer, and H. Walach, Found. Phys. 32, 379 (2002).
- Also note that the commutation relation for the x and y
angular momentum is [Lx,Ly]=iℏ Lz, i.e. not exactly
iℏ, but nobody is suggesting that this is therefore “weak” quantum
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