lördag 16 juli 2011

Monstrosity of Quantum Mechanics 7: Basic Postulates

In what sense are the basic postulates of quantum mechanics not Harry Potter fantasy?

Lubos Motl makes in The Unbreakable Postulates of Quantum Mechanics a heroic effort to justify quantum mechanics almost 100 years after its formulation, starting with:
  • Many people try to use every opportunity to diminish the importance of the principles of quantum mechanics and to cover them by fog and shadows - even though they're the most important and most robustly established insights of the 20th century science.
The mission is to convince skeptics about the truths of the following basic postulates:
  1. The set of possibilities in which a physical system may be found is described by a linear Hilbert space (more precisely by the rays in this space) equipped with an inner product.
  2. Complex (nonzero) linear combinations of allowed states are allowed states, too.
  3. A physical system composed out of N separated (or fully independent) subsystems has the Hilbert space equal to the tensor product of the Hilbert space describing the individual subsystems.
  4. Physical quantities, also referred to as "observables" in the fancy quantum mechanical context, are encoded in Hermitean (linear) operators acting on the Hilbert space.
  5. In particular, the evolution in time is generated by the operator known as the Hamiltonian.
  6. The exponentials of its imaginary multiples are the operators that evolve the system over a finite interval and these operators are unitary; similarly, other symmetry transformations are given by other unitary (or anti-unitary, if the time reversal is included) operators.
  7. The expectation values of the quantity "A" are given by the inner product ; if "A" is replaced by the projection operator "P", this expectation value expresses the probability that the condition connected with "P" will be satisfied once the system is measured.
The motivations for 1 - 7 presented by Lubos tell us something essential about the solidity of quantum mechanics. Let see how Lubos motivates 1 - 3:
  1. Why do we know that there is a Hilbert space? If a physical theory has a content, it must be able to manipulate with the information. We insert some information that we know - and it spits out another piece of information that we didn't know but that is predicted, or postdicted, by the theory. So there must exist some states; which state was realized in Nature, is realized in Nature, or will be realized in Nature, is the way to phrase all the information we have or we want to have about the world or its pieces. That was true even in classical physics: different states of a physical system were given by points in the phase space (spanned by the positions and their canonical momenta).
  2. The new thing about quantum mechanics is that the complex linear superpositions of two allowed states are also allowed states. How do we know that? Well, we may actually design procedures that create such combined states in practice.
  3. Now, there are other postulates and universal rules of quantum mechanics. For example, the composite systems are described by tensor products of Hilbert spaces. It's not hard to see why: if the dimensions of Hilbert spaces H1, H2 are equal to d1, d2, there are clearly d1 basis vectors of H1 and d2 basis vectors of H2. These basic vectors parameterize some linearly independent (i.e. fully mutually exclusive) possibilities. The set of linearly independent possibilities for the composite system obviously has to be the Cartesian product of the two sets for the separate subsystems. And the "linear envelope" of this Cartesian product - the new basis - is the tensor product of the original spaces. Its dimension - its number of basis vectors - is equal to d1.d2 as expected. This conclusion is pretty much inevitable, by basic logic.
When you read this as a mathematician you understand that the motivation is weak, formal and touches triviality elevated to deep insight into the true inner mechanisms of the microscopic world. The Hilbert space assumption essentially reflects that the Schrödinger equation is linear. But why physics on atomic scales is linear allowing superposition, is not motivated. This appears as an ad hoc assumption which could be made by one who has recently fallen in love with linear Hilbert space theory and has been so overwhelmed by emotion that rational thinking has disappeared.

The argument that "we may actually design procedures that create such combined states (superposed) in practice" sounds hollow, knowing that this principle of quantum computing has shown to be very difficult to demonstrate.

Atomic physics concerns the interaction of elementary particles by certain forces and thus can be thought as N-body problems. But an N-body problem is not linear, and so it requires a lot of fantasy to believe that the N-body problem of quantum mechanics through some miracle decides to show up as linear.

without being able to find any reasonable one.

2 kommentarer:

  1. Seen this?


  2. Yes. My position is that it is impossible to say if doubled CO2 will have a warming or cooling effect, but one can give several arguments indicating that the effect will be small, e.g smaller than plus minus 0.3C. In this sense I am a lukewarmist; I do not say, like some Slayers do, that the atmosphere has a cooling effect. I say that it has warming-cooling effect and that CO2 climate sensitivity is most likely so small that it can be forgotten.