Saturday, October 19, 2013

What is the number system of quantum mechanics?

The amazing power of dimensional analysis

Hardy’s 5 reasonable axioms paper was generalized by Jochen
Rau using dimensional analysis. To understand it we need to start with a
mathematical detour.

In the 1500s, it was fashionable to hold contests of solving
polynomial equations and this triggered the interest into finding general
solutions. Tartaglia found the solution for cubic equations, and Ferrari
discovered the solution for the quartic equation. Since no general solution was
found for the quintic equation, proving the impossibility for degree five and
above became a research topic.

The solution was found by Galois and the key was in the permutation of solutions. This lead to what is now
called the Galois group. This marriage between group theory and algebraic
equations led Sophus Lie (by the way he was Norwegian not Chinese) to wonder if something similar would
hold for continuous groups and differential equations. This idea was the
starting point of the Lie groups which are continuous
groups. As an example, the group of rotations in the ordinary three dimensional
space is a Lie group.

What is important is that each of this series has a definite
dimension. As a useful side note, the tangent space at the origin of any Lie
group forms a Lie algebra and this plays a major role in quantum mechanics.

Fast forward to present day, Jochen Rau considered the
following problem: why is a Lorentzian manifold distinguished from all other
event manifolds http://arxiv.org/pdf/1009.5523.pdf
? The solution combines physical principles with the mathematical fact of Lie
group dimensionality.

So if this method worked for orthogonal groups and
relativity, would it work for unitary groups and quantum mechanics? (The unitary group U(n) dimension is n^2)

The answer is yes and it resulted into a paper called
“Consistent reasoning about a continuum of hypotheses on the basis of finite evidence” .
While the paper title does not do justice to its content, the result is very
important and it represents a generalization of Hardy’s 5 reasonable axiom
result. In particular it clarifies what happens when r>2 for Hardy’s K=N^r
equation discussed in the prior post. When r=3 and above, probabilities are no
longer continuous. What does this mean?

Let us recall the most general relationship:

K = N a + 1/2! N (N-1) b + 1/3! N (N-1) (N-2) c + …

At each term there is new physics coming into play. The “a” terms correspond to old fashion
classical physics and no “spooky action at a distance”. The “b” terms correspond to quantum
superposition as seen from the |mn> element:

|mn> = 1/sqrt(2) (|m> + |n>)

What are the “c”
terms corresponding to? The “c”
terms correspond to new physics, must involve 3 wavefunctions |m>, |n>,
|p> which combines in such a way that it
is not a superpositionand the probabilities are not continuous
according to Rau’s result. Do we know such an object?

Yes we do, they are Feynman diagrams!!!

And the higher order terms correspond to Feynman
interactions where 4, 5, 6, n legs joined in one vertex. For each leg, there is a
discontinuity at the vertex, and the legs do not enter into a superposition.

Now the question becomes: can we find a sequence of numbers
a,b,c,d,… such that K = N^r with r>2? For now the answer is no, but the
problem is still open.

Next time I will start talking about quaternionic quantum
mechanics and the meaning of its lack of a tensor product.