On The Ether Concept In Physics

I don't know any complete theory of representation in infinite dimensional spaces, but given a Lie group acting on space-time, certainly you can find some convenient representations over infinite dimensional spaces. Suppose \(M\) is a manifold, and Lie group G acts on it \(G\times M\rightarrow M\). Now, take the space \(C^\infty(M)\) of smooth functions over M. If \(u\in C^\infty(M)\) and \(g\in G\), then define the action \((g,u)\mapsto g\cdot u\in C^\infty(M)\) by

\([g\cdot u](x)=u(g^{-1}x)\).

This is a representation over \(C^\infty(M)\), which is an infinite dimensional space.

Edit: This defines an action of G on \(C^\infty(M)\), but I don't know if representation means the same thing.

 

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Whilst I don't disagree with anything you've written, I don't see how it helps. Your example is almost a cheat - your action is pretty much defined by what it does on a finite dimensional vector space!

For example, let \(G=SU(N)\) and set \(V=\ell^p\) - I don't see a natural direction to go, and that's before I start worrying about issues of convergence!

It can't be simple, especially if it's to be defined on a vector space with additional structure: i.e in this case a Hilbert space for physics. Hmmm... interesting, perhaps one day I'll have to read about this!!

 

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But a Hilbert space for physics is always a function space over some manifold, isn't it?

 

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Sure thing - but I'm not a physicist!

Having just checked, he proves Wigner's theorem - regarding the existence of linear and unitary transformations on state space which correspond to symmetry transformations. I guess this would be the thing relevant to your post, although a small caveat is needed since the Lorentz group isn't simply connected.

Well that was a nice learning experience.

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I am reading Haag's book and I found the mention of Wigner's theorem there too. The caveat you are referring to is that you have to take the double cover of the Lorentz group right? I am just beginner but it seems that the double cover is in some sense more fundamental than the Lorentz group itself since from there you have the Dirac equation.

 

Notes by ScienceWeek:

In the late 19th century, what we now call "classical" physics incorporated the assumed existence of the "ether", a hypothetical medium believed to be necessary to support the propagation of electromagnetic radiation. The famous *Michelson-Morley experiment of 1887 was interpreted as demonstrating the nonexistence of the ether, and this experiment became a significant prelude to the subsequent formulation of Einstein's *special theory of relativity. Although it is often stated outside the physics community that the ether concept was abandoned after the Michelson-Morley experiment, this is not quite true, since the classical ether concept has been essentially reformulated into several modern *field concepts.

The following points are made by Frank Wilczek (Physics Today January 1999):

1) Isaac Newton (1642-1727) believed in a continuous medium filling all space, but his equations did not require any such medium, and by the early 19th century the generally accepted ideal for fundamental physical theory was to discover mathematical equations for forces between indestructible atoms moving through empty space.

2) It was Michael Faraday (1791-1867) who revived the idea that space was filled with a medium having physical effects in itself... To summarize Faraday's results, James Clerk Maxwell (1831-1879) adapted and developed the mathematics used to describe fluids and elastic solids, and Maxwell postulated an elaborate mechanical model of electrical and magnetic fields.

3) The achievement of Einstein (1879-1955) in his paper on special relativity was to highlight and interpret the hidden symmetry of Maxwell's equations, not to change them. The Faraday-Maxwell concept of electric and magnetic fields, as media or ethers filling all space, was retained by Einstein. Later, Einstein was dissatisfied with the particle-field dualism inherent in the early atomic theory, and Einstein sought, without success, a unified field theory in which all fundamental particles would emerge as special solutions to the field equations.

4) Following Einstein, Paul Dirac (1902-1984) then showed that photons emerged as a logical consequence of applying the rules of quantum mechanics to Maxwell's electromagnetic ether. This connection was soon generalized so that particles of any sort could be represented as the small-amplitude excitations of quantum fields. Electrons, for example, can be regarded as excitations of an electron field, an ether that pervades all space and time uniformly. Our current and extremely successful theories of the *strong, electromagnetic, and weak forces are formulated as *relativistic quantum field theories with *local interactions.

5) The author states: "Einstein first purified, and then enthroned, the ether concept. As the 20th century has progressed, its role in fundamental physics has only expanded. At present, renamed and thinly disguised, it dominates the accepted laws of physics."

Science Week

Physics Today

cinci: An interesting article and one that shouldn't offend relativists or anti-relativists. It seems to me the whole thing would be a lot clearer if the modern version of ether were given different name....I vote for "quether", short for quantum ether. Ether is dead....long live quether!

 

I don't know about more fundamental, but certainly significant. The double-cover aspect encapsulates (mathematically) the notion of spin.

 

Actually, to add to that:

I first came across spinors in a QFT course, in which it made little sense to me to define something by means of its behavior under Lorentz transformations. With hindsight, this view is clear - but I found the most enlightening (and elementary) view was that of Cartan's. Whilst it's not mathematically fancy, it's solid and gives a firm basis for the use of spinors, and then onto the spin groups. It's all about isotropic vectors in C^n. An excellent and cheap account is given in Cartan's "Spinors".

It seems the modern way to introduce spinors is abstract and complicated, but upon reading this book you'd think it was an undergrad topic.