(pg. 178-84, Quantum Theory By David Bohm)

I'll have to build up to it...

  "A basic requirement that any Phi must satisfy is that it be quadratically integrable, i.e., that

\integral (from -\inf to +\inf)|Phi|^2dx = a finite quantity."

[in natural language: the combination of all of the infinitesimal parts that range from negative to positive infinity of the square of the absolute value of the wave function given a change in x equals some finite number. (? Is this correct ?) This is a necessary condition for the normalization of the probabilities.]

1) What is dx (or dp, for that matter)? Infinitesimal changes in x or p?

2) Why is it assumed that all probabilities are equally likely? (Principle of Insufficient Reason?)

There is a discussion of how "averages of all physical observable quantities must exist": "... x and p are clearly physically observable properties, so that there averages must exist."

3) So *all* "averages" exist ontologically?

4) "Where"? In some Hilbert space (or Gelfand triple)? (http://xxx.lanl.gov/abs/quant-ph/9712038)

Continuing: "...whenever we know a given function is physically important, we require of all acceptable wave functions that the average value of this quantity shall exist."

5) Can we think of this as saying: "If I can observe a physical property, I require that of all acceptable wave functions, given my local standard, that the average value of this quantity shall exist"?

6) What about an infinity of observers, each with their own "version" of 5), are the "averages" of one necessarily be the same as another's; how would it be possible to know this since observers can not compare their local gauges of time and distance to each other? ("no connections between LSs")

Bohm goes on to say that the method of evaluating the average value of any function of x and of p is done using an operator to represent one of these quantities with an operator, e.g. if we wish to use a co-ordinate representation we say p = hbar/i\partial/\partial x and if we wish to use a momentum representation we use x = ihbar\partial/\partial p; and plug these into our integration equation ranging from negative to positive infinity as x or p shift infinitesimally.

Bohm then relates the "classical limit" to the "correspondence principle": "To show that it [the operator] does satisfy the correspondence principle[1], we consider a wave function which takes the form of a wave packet ["wavelet"?]. Insofar as all classical results are concerned, no important physical quality can change appreciably [2] within the packet [3]. This is because, in the classical limit, the packet looks essentially like a particle [3!], so that if the system is to be described classically, the specific wave-like properties of the packet must not matter [3!]. Hence, we may neglect all changes of x within the packet, and replace x with x^\average, which may be regarded as essentially constant."...

"Although the above gives the correct classical limit for averages of f(x,p), it is somewhat ambiguous because the order in which the operators x and p appear is vital, whereas in the corresponding classical expression, this order is immaterial.[4] We now shall that this ambiguity is removed in part by the requirement that the mean value of any real function of x and p must be real for an arbitrary Phi."...

"it is easy to show, for example, that xp^\average as defined above is not real. To do this we write

xp^\average = \integral(from -\inf. to + \inf.) Phi*x hbar/i \part.Phi/\part.x dx (11a)

Integrating by parts yields (noting that the integrated part vanishes)

xp^\average = - hbar/i \integral(from -\inf. to + \inf.) Phi*x hbar/i \part.Phi/\part.x dx

= - hbar\integral(from -\inf. to + \inf.)(Phi*Phi + Phi(x)(\part.Phi*/\part.x))dx (11b)

we note that the second term of the right-hand side of the above expression is equal to the complex conjugate of xp^\average. Hence xp^\average is equal to its complex conjugate plus an additional term; this means that xp^\average cannot be real [valued].

16. Hermite operators: To avoid such complex [valued] averages for quantities which are basically real [when multiplied by their conjugate, which it its 'time inverse'!], we shall require, as has already been stated, that the mean value be defined such that it is real for arbitrary Phi. if O(p,x) is the operator in question, we require the O^\average be equal to its complex conjugate [5]. Now we have

O^\average = \integral(from -\inf. to + \inf.) Phi*(x)O Phi(x) dx (12)

the complex conjugate of O^\average is found by taking the complex conjugate of all parts of the integral. Hence the reality requirement is equivalent to the following:

\integral(from -\inf. to + \inf.)Phi*(x)OPhi(x)dx = \integral(from -\inf. to + \inf.)Phi*(x)O*Phi*(x)dx (13)

O* refers to the complex conjugate of the operator O. For example in the operator p = hbar/i \part./\part.x, we get p* = - hbar/i \part./\part.x. Operators satisfying equation 13 are said to be Hermitian."

1 ["that the laws of quantum physics must be so chosen that in the classical limit, where many quanta are involved, the quantum laws lead to the classical equation as an average" pg. 31 ibid.]

2 [more than an infinitesimal amount?]

3 [this looks like a comment that Robert Fung made about LSs looking like LSs!]

4 [the "geometry" of event relations is anti-commutative in QM and commutative in Newton's model?]

5 ["equal" or "equivalent set-wise"?


from Jeeva Anandan's paper "Quantum Measurement Problem and the Gravitational Field" in The Geometric Universe, edited by S. A. Huggett et al; pg. 359

"...the state vector undergoes two types of changes, which using the terminology of Penrose, may be called the U and R process. ... The U process is the linear unitary evolution which in the present day quantum theory is governed by Schroedinger equation. But what causes the measurement problem is the linearity of the U process. The unitarity is really relevant to the R process. Unitarity ensures that the sum of the probabilities of the possible outcomes in any measurement, each of which is given by an R process remains constant during the U time evolution. This of course follows from the postulate that the transition probability from the initial to the final state in the R process is the square of the modulus of the inner product between the normalized state vectors representing the two states.

The process of measurement ... takes place in two stages: first is the entanglement and second is the collapse. If we had no choice in preparing the initial state [beware of "initiality"!] of the system then Phi would be in general a superposition of the Phi_i's. Then the entanglement ... would be inevitable consequence of the linearity of the evolution. But if we could prepare the state then it is possible to prevent entanglement as in the case of protective observations ( ) That is, in such an observation

Phi alpha -> Phi' alpha' (2.4)

where the state represented by the Phi' does not differ appreciably from the state Phi. The protection is usually an external interaction which puts Phi in an eigenstate of the Hamiltonian and the measurement process results in adiabatic ["occurring without loss or gain of heat"] evolution.

Then alpha' gives information about Phi; specifically it tells us the 'expectation value' with respect to Phi of the observable of the system that is coupled to an apparatus observable. By doing such experiments a large number of times it is possible to determine Phi (up to phase) even though the system is always undergoing U evolution. Concequently, the statistical interpretation of quantum mechanics is avoided during protective measurements. Indeed, Phi may be determined using just one system which is subject to many experiments."