In 1939, Paul Dirac observed that “the physicist, in his study of natural phenomena, has two methods of making progress”: experiment and observation, and mathematical reasoning. Although he said, “there is no logical reason why the second method should be possible,” nevertheless it works, and to great effect. The key, Dirac felt, was beauty, leading him to his principle that successive theories of nature are characterized by increasing mathematical beauty. The results of this were rich and included some predictions not confirmed until after Dirac’s death. Nevertheless, the powerful guidance Dirac found in mathematics did sometimes lead him astray, as he rejected the principle of “renormalization,” developed by Feynman, Schwinger, and Tomonaga, to remedy the nonphysical infinities that kept cropping up in Dirac’s equations for quantum electrodynamics. Even as other physicists accepted it, Dirac never did, saying it was “just not sensible mathematics.” Nevertheless, it was powerful physics.He followed with this interview:

Why do the atoms have those properties? Because they’re made of quarks and electrons. What about the electron? What properties does it have? And the cool thing is, all the properties that electrons have are purely mathematical. It’s just a list of numbers. So in that sense, an electron is a purely mathematical object. In fact, there’s no evidence right now that there’s anything at all in our universe that is not mathematical.Yes, there is evidence. There is no purely mathematical description of an electron because, according to the rules of quantum mechanics, measurements on an electron can depend on measurements of a possibly-distant entangled electron.

The usual explanation is that the two electrons have a joint mathematical description as a spinor wave function in a tensor product Hilbert space. This leads to paradoxes of relativity and causality.

There are different interpretations of quantum mechanics. The electron is never just a list of numbers. All attempts to reduce the electron to some lcoal set of numbers have failed. We can predict measurements of an alectron based on previous measurement of that electron as well as any entangled electrons. But we cannot reduce the electron to numbers.

Dirac got Einstein's disease, and was unproductive once he got the idea that physical theories could be predicted by mathematical beauty.

Tegmark says he is writing a book, and will promote it on his Facebook page.

"we cannot reduce the electron to numbers."

ReplyDeleteDo you agree that the physical world is well described by quantum mechanics? Well, if you do, then you will agree that the quantum state of the universe is given by an element of a separable complex Hilbert space. But any infinite dimensional separable complex Hilbert space is unitarily equivalent to l^2, so the quantum state of the universe (and thus the state of any electron) can be represented by a list of complex numbers.

Q.E.D.

I agree that quantum mechanics well describes electron observations, and that all those Hilbert spaces are equivalent to l^2. But if that were the whole story, why do people keep looking for hidden variables?

ReplyDeletePerhaps both electrons in a helium atom can be properly jointly represented by a wave function, but each individual electron cannot be.

"But if that were the whole story, why do people keep looking for hidden variables?"

ReplyDeleteBeats me.

"Perhaps both electrons in a helium atom can be properly jointly represented by a wave function, but each individual electron cannot be."

Actually it can. It's not possible to solve Schrodinger's equation analytically in the case of a helium atom, but we can find the approximate wavefunction of one of the electrons. The answer is given on p.212 of Griffiths' book.

Dear Roger,

ReplyDeleteReality as some kind of a mathematical structure was hinted at by many scientists before Tegmark, like Conway, Wolfram, Wheeler and others. Tegmark just worded the idea in a very bold and direct way. The traditional view that reality is some "physical" substance (or some unknown something)has impossible implication. where did this substance come from and where does it reside? and both create a perpetual paradox. a mathematical structure like a circle does not need more elementary construct or a place to reside, so is our reality.

If nothing existed where would mathematical facts exist, I don't know why people have a hard time with this question which has a very obvious answer.

Reality exists hence we say it is true. But what is really true besides that more than anything else which we can really trust, it is mathematical facts. So, to my mind I connect both since both seem to be a statement of truth. So I took a guess that reality is something akin to a circle (truth). The relations between the points give you a mathematical structure whereby you get PI which defines the structure of the circle.

the structure that leads to our reality is random numbers and certain unavoidable relations(and only possible ones) between them. that is all. It is the only dynamic design possible with fundamental entities, all other designs lead to a static or quasi-static.

http://www.qsa.netne.net/