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Dirac's "Lectures on QFT" (1964-65) Lecture 8

Here are some quotes from Dirac, which justify the search for a deeper and more logically adequate quantum theory. From p.1:

This work though has serious difficulties in it. There are many qualities appearing in the theory which turn out to be infinite although they ought to be finite; and people have followed all sorts of tricks for avoiding these difficulties, but the result is that the theory is in rather a mess. I ought to confess that I really don't understand it myself. The departures from logic are very serious and one really gives up all pretense of logical development in places...

from p.3:

People usually say that these equations establish the equivalence of the two pictures, that one can use whichever one likes indiscriminately. However, the argument is valid only provided e^iHt exists as an operator in Hilbert space and for the Hamiltonians which one meets with in quantum field theory there is good reason to believe that this is not so and the two pictures are not equivalent...

from p.5:

For simple examples, where we have only a finite number of degrees of freedom, the operator e^iHt does exist alright and all that you have learnt can still be retained; but for quantum field theory the two pictures are not equivalent and one must think of the Heisenberg picture as the more fundamental one which is valid in nature...

from p.6:

It is not possible to get a solution of the Schrodinger equation for which the ket vector stays in Hilbert space...

from p.9:

We cannot get solutions in the Schrodinger picture with the kets remaining in Hilbert space, and corresponding to that we cannot get solutions of the Heisenberg picture in which our q-numbers operate only on the kets of a Hilbert space. We have to imagine that the q-numbers of the Heisenberg picture operate on the vectors of some space which is bigger than a Hilbert space. I don't know the mathematical nature of this more general space--it is better to leave it unspecified for the present rather than to make a guess which might later turn out to be unsuitable for physical purposes. I expect that mathematicians will make guesses of what the space is, but they might very well guess wrong and they already have made a wrong guess when they thought that Hilbert spaces were adequate for physical purposes...

So, the point is that without even discussing measurement theory, there are already major contradictions and problems with conventional quantum theory. There seems to be something wrong with the logical structure of theory - it does not seem to be logically self-consistent. When it comes to measurement theory, it is not even clear that it is coherent.

Therefore, there is plenty of room for improvement and for seeking to discover a new and logically superior foundation for the theory. I suspect that it is simple Aristotelian logic (in the context of Platonic ontology) that is missing from the theory, and this is due to the painful divorce of physics from philosophy.

Fragmented people think fragmented thoughts that never add up and ultimately don't make sense, whatever illicit success may initially obtain. The deepest urge in my being is to understand the principles of physics from the first principles of logic. Without that our claim to knowledge is nought but a pretense, and our souls are divided against themselves, leaving us culturally schizophrenic and socially insane.

Peter Joseph Mutnick 1949 - 2000