algebras are also useful in statistical mechanics. Actually there's an interesting feedback loop here: ideas from statistical mechanics have also had an important effect on the theory of C*-algebras. The most famous example is the work of Kubo, Martin and Schwinger on thermal equilibrium, which led to something called Tomita-Takesaki theory. But I said I wouldn't get too mathematical, so I can't talk about this!
I've already hinted that there's a close relation between "vacuum states" and representations of C*-algebras. In fact there's a theorem called the GNS construction which makes this precise. But physically, what's going on here? Well, it turns out that to define the concept of "vacuum" in a quantum field theory we need more than the C*-algebra of observables: we need to know the particular representation. This becomes most dramatic in the case of quantum field theory on curved spacetime - a warmup for full-fledged quantum gravity. It turns out that in this setting, it's a lot harder to get observers to agree on what counts as the vacuum than it was in flat spacetime. The most dramatic example is the Hawking radiation produced by a black hole. You may have heard pop explanations of this in terms of virtual particles, but if you dig into the math, you'll find that it's really a bit more subtle than that. Crudely speaking, it's caused by the fact that in curved spacetime, different observers can have different notions of what counts as the vacuum! And to really understand this, C*-algebras are very handy.
Well, this article is so lacking in detail that I'm getting sort of sick of writing it - the really fun part, to me, is how the mathematics of C*-algebras makes the vague verbiage above utterly precise and clear! So I'll stop here. If you want some details, try these books:
Gerard G. Emch, Algebraic Methods in Statistical Mechanics and Quantum Field Theory, Wiley-Interscience, New York, 1972.
Rudolf Haag, Local Quantum Physics: Fields, Particles, Algebras, Springer, Berlin, 1992.
Ola Bratteli and Derek W. Robinson, Operator Algebras and Quantum Statistical Mechanics, 2 volumes, Springer, Berlin, 1987-1997.
For books that talk about the math but not the physics, try these (in rough order of increasing difficulty):
William Arveson, An Invitation to C*-Algebras, Springer, New York, 1976.
Masamichi Takesaki, Theory of Operator Algebras I, Springer, Berlin, 1979.
Richard V. Kadison and John R. Ringrose, Fundamentals of the Theory of Operator Algebras, 4 volumes, Academic Press, New York, 1983-1992.
Shoichiro Sakai, C*-Algebras and W*-Algebras, Springer, Berlin, 1971.
I've already hinted that there's a close relation between "vacuum states" and representations of C*-algebras. In fact there's a theorem called the GNS construction which makes this precise. But physically, what's going on here? Well, it turns out that to define the concept of "vacuum" in a quantum field theory we need more than the C*-algebra of observables: we need to know the particular representation. This becomes most dramatic in the case of quantum field theory on curved spacetime - a warmup for full-fledged quantum gravity. It turns out that in this setting, it's a lot harder to get observers to agree on what counts as the vacuum than it was in flat spacetime. The most dramatic example is the Hawking radiation produced by a black hole. You may have heard pop explanations of this in terms of virtual particles, but if you dig into the math, you'll find that it's really a bit more subtle than that. Crudely speaking, it's caused by the fact that in curved spacetime, different observers can have different notions of what counts as the vacuum! And to really understand this, C*-algebras are very handy.
Well, this article is so lacking in detail that I'm getting sort of sick of writing it - the really fun part, to me, is how the mathematics of C*-algebras makes the vague verbiage above utterly precise and clear! So I'll stop here. If you want some details, try these books:
Gerard G. Emch, Algebraic Methods in Statistical Mechanics and Quantum Field Theory, Wiley-Interscience, New York, 1972.
Rudolf Haag, Local Quantum Physics: Fields, Particles, Algebras, Springer, Berlin, 1992.
Ola Bratteli and Derek W. Robinson, Operator Algebras and Quantum Statistical Mechanics, 2 volumes, Springer, Berlin, 1987-1997.
For books that talk about the math but not the physics, try these (in rough order of increasing difficulty):
William Arveson, An Invitation to C*-Algebras, Springer, New York, 1976.
Masamichi Takesaki, Theory of Operator Algebras I, Springer, Berlin, 1979.
Richard V. Kadison and John R. Ringrose, Fundamentals of the Theory of Operator Algebras, 4 volumes, Academic Press, New York, 1983-1992.
Shoichiro Sakai, C*-Algebras and W*-Algebras, Springer, Berlin, 1971.
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