Paul P. Mealing

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Wednesday, 1 June 2011

Quantum Platonism

This post is a logical extension of the previous one – a sequel if you like – and, for that reason, it should be read in conjunction with it.

One of the things I learnt, from researching for that post, was that Schrodinger was attempting something else to what he achieved. He didn’t like the consequences of his own equation. I believe he was expecting to obtain results that would reconcile quantum phenomena with classical physics and that didn’t happen. His famous Schrodinger’s Cat thought experiment confirms his disbelief in Bohr’s and Heisenberg’s interpretation of the wave function collapse: only when someone makes an observation or a measurement does reality occur. Prior to this interaction, the quantum state exists as a superposition of states simultaneously. His thought experiment was to take a quantum phenomenon and amplify it to a contradictory macro-state: a cat that was dead and alive at the same time. His express purpose was to illustrate how absurd this was.

Likewise, he apparently wasn’t happy with Born’s probabilities, yet it was Born’s insightful contribution that actually gave Schrodinger what he wanted: a connection between his quantum wave function and classical physics. To quote Arthur I. Miller in Graham Farmelo’s book, It Must be Beautiful; Great Equations in Modern Science:

[Born’s] dramatic assumption transformed Schrodinger’s equation into a radically new form, never before contemplated. Whereas Newton’s equation of motion yields the special position of a system at any time, Schrodinger’s produces a wave function from which a probability can easily be calculated… Born’s aim was nothing less striking than to associate Schrodinger’s wave function with the presence of matter. (My emphasis)

I think this is the key point: Born was able to provide a mathematical connection between quantum physics and classical physics via probabilities. The fact that these probabilities agreed with experimental data is what cast Schrodinger’s equation in stone and gave it the iconic status it still has in the 21st Century. As Wikipedia points out: Schrödinger's equation can be mathematically transformed into Richard Feynman's path integral formulation, which is the basis of his QED (quantum electrodynamics) analytic method, and the current ‘last word’ on quantum mechanics.

I re-read Feynman’s ‘lectures’ on QED after writing my post and one can see the connection clearly. But it’s Born’s influence that one sees, rather than Schrodinger’s, which is not to diminish Schrodinger’s genius. His attempt to create a ‘visualisable’ wave function, as opposed to Heisenberg’s matrices, is what set the course in quantum mechanics for the rest of the century.

But whilst Schrodinger and Einstein argued over the philosophical consequences of quantum mechanics with Bohr and Heisenberg, Feynman (a generation later) was dismissive of philosophical considerations altogether. In a footnote in QED, Feynman argues that the probability amplitudes are all that matters, and that the student should ‘avoid being confused by things such as the “reduction of a wave packet” and similar magic.’

If Feynman professes a philosophy it is by this credo:

‘I’m going to describe to you how nature is – and if you don’t like it, that’s going to get in the way of your understanding it… So I hope you can accept Nature as She is – absurd.’

However, the discontinuity between quantum mechanics and classical mechanics that arises from a ‘measurement’ or an ‘observation’ is hard to avoid. As I said in my previous post, it is entailed in Schrodinger’s equation itself, because the wave function is continuous yet all quantum phenomena are discrete. Roger Penrose, and others (like Elwes, quoted in previous post) point out that Schrodinger’s wave function is continuous until the quantum phenomenon in question is physically resolved (observed), whence the wave function effectively disappears.

What this tells me is that everything seems to be connected. It’s like nothing can come into existence until it interacts with something else. But it also implies that the quantum world and the classical world – what we call reality – are distinct yet interconnected. It reminds me of Plato’s cave, where our reality is akin to the ‘shadows’ projected from a quantum world that only mathematics can describe with any precision or purpose.

Our reality is a veneer and the quantum world hints at a substratum that obeys different rules yet dictates our world. It’s only through mathematics that we are able to perceive that world let alone comprehend it – particle smashers play their role, but they only provide windows of opportunity rather than a panoramic view.

This is a subtly different concept to the ‘hidden variables’ philosophy proposed by David Bohm (and some say Einstein) because I’m suggesting that the quantum world and the classical physical world obey different rules.

In a not-so-recent issue of New Scientist (30 April 2011, pp.28-31) Anil Ananthaswamy explains how different parties (Mario Berta from the Swiss Federal Institute of Technology, Robert Prevedel of the University of Waterloo Canada and Chuan-Feng Li of the University of Science and Technology of China in Hefie) have all reduced the limits of Heisenberg’s uncertainty principle through quantum entanglement.

Their efforts were apparently in response to theoretical suggestions by 2 Dutch physicists, Hans Maassen and Jos Uffink, that information gained through quantum entanglement (knowing information about one entangled particle or photon axiomatically provides information about its partner) would affect the limits of Heisenberg’s uncertainty principle. For example: if 2 particles go in opposite directions after a collision, they theoretically have the same momentum, yet Heisenberg’s uncertainty principle states that the information would be necessarily fuzzy, juxtapose knowing its position. However, measuring the momentum of one particle automatically gives knowledge of the other that subverts the uncertainty principle for the second particle.

Entanglement is an example of quantum interaction that classical physics can’t explain or even duplicate. That there appears to be a correspondence between this and the uncertainty principle supports the view that the quantum world obeys its own rules.

In my introduction, I suggested that this post needs to be read in conjunction with the previous one. This post focuses on the philosophy of quantum mechanics whereas the previous one focused on the science. Whereas the philosophy of quantum mechanics is contentious, the science is not contentious at all. That’s why it’s important to appreciate the distinction.

11 comments:

Anonymous said...

Hi, I am from Melbourne too.

Please find a unique understanding of Quantum Reality via these references.

Reality & The Middle via:

www.dabase.org/s-atruth.htm

Broken Light

www.dabase.org/broken.htm

Einstein meets Jesus

www.dabase.org/christmc2.htm

Artists Statement

www.aboutadidam.org/readings/transcending_the_camera/index.html

The Truth About Science (and scientism)

www.adidam.org/teaching/aletheon/truth-science.aspx

The Baneful Limitations of Scientism

www.aboutadidam.org/lesser_alternatives/scientific_materialism/index.html

Plus Art & Physics by a professor of surgery

www.artandphysics.com

Paul P. Mealing said...

Hi Anonymous,

I had a quick perusal of your links, of which only a couple appear to be relevant to science at all. I have to admit that my first impression of Avatar Adi Da Samraj is not to be impressed.

People who mix science and religion often have a superficial knowledge of the science and an over-inflated opinion of their own erudition. Unfortunately, that's the impression I get from Avatar Adi Da Samraj.

Science and religion deal with different things. Science is the study of natural phenomena in all its manifestations. Religion is about a person's inner journey: their quest for meaning and a life's purpose.

One is concerned with the outer world, one with the inner world; one is objective the other totally subjective. Epistemologically, they couldn't be further apart.

The only bridge between them is mathematics, because mathematics only becomes manifest as an abstract realm and a tool of perception, in the mind of a sentient being.

Mathematics is what has made science the most successful human enterprise ever, for 2 reasons: the extraordinary concordance between mathematics and the natural world; and because mathematics provides the only 'universal truths' we can truly depend on.

Regards, Paul.

David Edwards said...

Paul,

The Feynman Integral wasn't Feynman's basis for QED because that would require Super Feynman Integrals, and Feynman wasn't able to understand how to handle Fermi fields with his integrals; that was done later by others. Feynman 'just' guessed the correct perturbation expansion; an expansion which is known not to converge and for which there is no proof that it is an asymptotic expansion for the S-matrix. There is a $1,000,000 prize for making QED and other gauge theories such as the standard model mathematically well-defined.

Dave

Dr. David A. Edwards
Department of Mathematics
University of Georgia
Athens, Georgia 30602
http://www.math.uga.edu/~davide/
http://davidaedwards.tumblr.com/
dedwards@math.uga.edu

Paul P. Mealing said...

Hi Dave,

I always appreciate comments from someone who knows more than me.

Feynman 'just' guessed the correct perturbation expansion; an expansion which is known not to converge and for which there is no proof that it is an asymptotic expansion for the S-matrix.

I have to admit I don't really know what that means.

I'm currently reading Barrow's and Tipler's The Anthropic Cosmological Principle, and they point out that Feynman's 'sum-over-histories' method was based on the wave function and the 'action principle', which led to gauge theory, if I understand them correctly (p.152, Oxford University Press, 1986).

I also recently read a 'Masters' thesis by James C. Emerson at San Jose State University, who's developed a different Platonic interpretation of quantum mechanics, where he combines quantum information theory (QIT) with Bohmian hidden variables and suggests that the 'wave function collapse' is in fact 'Platonic Instantiation' (PI).

Thanks for taking an interest in my blog.

Regards, Paul.

David Edwards said...

Paul,

Try:

http://www.amazon.com/QED-Men-Made-Silvan-Schweber/dp/0691033277/ref=sr_1_1?ie=UTF8&qid=1310638947&sr=8-1

for a very interesting, readable exposition of the development of QED.

Two thoughts of mine:

1. The following is a program I'm trying to develop: a theory of interlocking worlds.

The central message that Bohr and von Neumann taught us about the Standard Quantum Logic is that it can be viewed as a manifold of interlocking perspectives that cannot be embedded into a single perspective. Hence, the perspectives cannot be viewed as perspectives on one real world. So, even considering one world as a methodological principle breaks down in the quantum micro-domain. The issue I'm pondering is the inadequacy of only talking about appearances and not going beyond appearances to some sort of world. Appearances are very complicated, confused, etc. Worlds are both simpler and more inclusive. I have no problem merely assuming some sort of world if it works! That is, simplifies our conceptions. Think of what happened to Chew's S-matrix approach; it lacked powerful enough heuristics to get anywhere.

To give you an idea of what I’m talking about concerning interlocking worlds consider the following definition of a quantum phase space associated to a quantum system described by a non-commutative C* algebra B. First replace B by its diagram of commutative sub-C* algebras {A}. Then apply the functor D which replaces a commutative C* algebra A with its maximal ideal space D(A)-a compact topological space-to {A} to get {D(A)}. This diagram is the quantum phase space of a quantum system. This is the type of gadget which could be described as interlocking worlds. Of course, in general, we’ll have to deal with more loosely, vaguely defined diagrams. The above could be a precise model of Cartwright's Dappled World. It is identical to Hawking's recent idea of model dependent realism.

2. The singularities of ordinary General Relativity can be avoided by considering the (mathematically well-defined) Einstein-Yang-Mills-Dirac system which is (heuristically) the super-classical limit of the (not mathematically well-defined) Standard Model. This system has complete solutions without singularities, solitons, and a Cyclic Universe solution. (The system has negative energy density; hence doesn't satisfy the positivity conditions in the Penrose-Hawking Singularity Theorems.) One would like to be able to apply deformation quantization to this system to get a mathematically rigorous Standard Model in a Curved Noncommutative Spacetime. This would obviate any necessity of going further to a quantum gravity theory. The deformation parameter would be Planck’s constant and the associated Planck Length would be the scale at which the manifold structure of spacetime gives way to a rougher structure of a noncommutative geometry. The Einstein-Yang-Mills-Dirac Equations provide an alternative approach to a Cyclic Universe which Penrose has recently been advocating. They also imply that the massive compact objects now classified as Black Holes are actually Quark Stars, possibly with event horizons, but without singularities.

Dave

Dr. David A. Edwards
Department of Mathematics
University of Georgia
Athens, Georgia 30602
http://www.math.uga.edu/~davide/
http://davidaedwards.tumblr.com/
dedwards@math.uga.edu

Paul P. Mealing said...

Hi Dave,

You are way over my head on this, but there are other people, who follow this blog, who probably can follow you.

If I could talk to you face to face, with a blackboard, I may have a small chance of keeping up.

I have read Penrose's book on cyclic universes and wrote a review on it here.

I checked out your link - it looks like one of those books I should read. I also looked up all of Jack Sarfatti's reviews, which took me off on another tangent: especially his review of Eternity to Here by Sean M. Carroll.

Regards, Paul.

David Edwards said...

Paul,

You might find it interesting that Platonism is the law of the land!

Why Shouldn't the Laws of Nature be Patentable

http://www.math.uga.edu/%7Edavide/Why_Shouldnt_the_Laws_of_Nature_be_Patentable.pdf

Dave

P.S. I think you might also find the following sociological facts interesting:

Every time I've made the following statement to mathematicians, they've responded with a broad grin and total agreement!

Even if God hadn't created the Universe, still there would be an infinite number of primes.

Recently, I followed the above with the following:

Even if God hadn't created the Universe, still there would be the category of categories.

I was amazed to discover that young (~30) mathematicians also totally agree with the second statement modulo logical qualms!!



--
Dr. David A. Edwards
Department of Mathematics
University of Georgia
Athens, Georgia 30602
http://www.math.uga.edu/~davide/
http://davidaedwards.tumblr.com/
dedwards@math.uga.edu

Paul P. Mealing said...

Hi Dave,

I love your quotes, especially the first one, because the second one went over my head. Are you referring to Langland's program?

Pity we can't talk one to one - I'm sure I could learn a lot from you.

Regards, Paul.

David Edwards said...

Paul,

The second quote just refers to category theory; see:

http://en.wikipedia.org/wiki/Category_theory

http://plato.stanford.edu/entries/category-theory/

You write: "Pity we can't talk one to one - I'm sure I could learn a lot from you." Email works well!

dedwards@math.uga.edu

I have plenty of time=I'm retired=I've attained Nirvana=I'm on full time R&R=Research&Racquetball!!

Dave

David Edwards said...

Paul,

The following is simultaneously the most profound and funniest insight into the mathematical experience:

The ideal mathematician

https://people.maths.ox.ac.uk/bui/ideal.pdf

Dave

Paul P. Mealing said...

Yes, real mathematicians are an elite. People like Marcus du Sautoy do an excellent job of demystifying it for the rest of us.

I'm currently reading The Great Equations by Robert P. Crease, which is a well written book for people like me.

Mathematics is a double whammy for most people, because, not only is the symbolism 'foreign', but so are the concepts that the symbols represent. If you can't grasp the concepts then you can't grasp the symbols, yet it's the symbols that people generally have to face on their first acquaintance.

Regards, Paul.