Timeline for Mathematical foundations of Quantum Field Theory
Current License: CC BY-SA 3.0
6 events
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| Jun 6, 2018 at 0:05 | comment | added | gradstudent | @Clay Cordova I am a bit curious about this definition of your Hilbert space in QFT as "it has a basis in correspondence with fields, i.e. functions from space to say $\mathbb{R}$". So I guess the quantum states of this QFT are now elements of this Hilbert space you defined, right? And are the quantum fields (need not be gauge invariant - can carry free gauge/colour indices, right?) now linear transformations of this Hilbert space onto itself? | |
| Apr 21, 2012 at 22:29 | comment | added | Theo Johnson-Freyd | @Sergei: Up to complex conjugation, a Hilbert space is equivalent to its dual. But often when giving non-technical discussion of physics structure, the word "Hilbert space" is used to mean something like "quantum phase space" or "vector space" or something, without the usual mathematical meaning to to the word "Hilbert". Note that @Clay's (good) answer is carefully non-specific about analytic issues like "what type of topology to use". But to answer your question: Clay's vector space $V$ comes with a distinguished basis, and hence a distinguished positive-definite pairing. | |
| Apr 16, 2012 at 7:01 | comment | added | Sergei Akbarov | This sounds strange: "The Hilbert space of the theory is then the dual space (over C) to V". Why the dual space must be Hilbert? | |
| Apr 16, 2012 at 1:45 | history | edited | Clay Cordova | CC BY-SA 3.0 |
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| Apr 16, 2012 at 1:30 | history | edited | Clay Cordova | CC BY-SA 3.0 |
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| Apr 16, 2012 at 0:58 | history | answered | Clay Cordova | CC BY-SA 3.0 |