advanced_tools:wick_rotation
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advanced_tools:wick_rotation [2017/06/14 10:31] jakobadmin [Why is it interesting?] |
advanced_tools:wick_rotation [2018/03/12 15:29] (current) jakobadmin |
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| <tabbox Student> | <tabbox Student> | ||
| - | <note tip> | + | <blockquote> |
| - | In this section things should be explained by analogy and with pictures and, if necessary, some formulas. | + | This is widely used to convert quantum mechanics problems into statistical mechanics problems by means of Wick rotation, which essentially means studying the unitary group exp(−itH/~) by studying the semigroup exp(−βH) and then analytically continuing β to imaginary values. |
| - | </note> | + | |
| + | <cite>https://arxiv.org/pdf/1311.0813.pdf</cite> | ||
| + | </blockquote> | ||
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| <tabbox Researcher> | <tabbox Researcher> | ||
| + | <blockquote>Unfortunately, relatively little is | ||
| + | known about Yang-Mills fields on Minkowski spacetime and, worse yet, the | ||
| + | basic objects of interest in quantum field theory (Feynman path integrals) | ||
| + | are extraordinarily difficult to make any sense of in this indefinite context. | ||
| + | The minus sign in the Minkowski inner product is rather troublesome. Not | ||
| + | to be deterred by such a minor inconvenience, the physicists do the only | ||
| + | reasonable thing under the circumstances—they change the sign! To lend | ||
| + | an air of respectability to this subterfuge, however, they give it a name. | ||
| + | Introducing an imaginary time coordinate τ = it is designated a Wick ro- | ||
| + | tation and has the laudable effect of transforming Minkowski spacetime into | ||
| + | R 4 (x 1 x 2 + y 1 y 2 + z 1 z 2 − t 1 t 2 = x 1 x 2 + y 1 y 2 + z 1 z 2 + τ 1 τ 2 ). What more could | ||
| + | you ask? Well, of course, a pedant might ask whether or not any physics | ||
| + | survives this transformation. This is a delicate issue and not one that we | ||
| + | are prepared to address. The answer would seem to be in the affirmative, | ||
| + | but the reader will have to consult the physics literature to learn why (see | ||
| + | Section 13.7 of [Guid]). Whether or not there is any physics in this positive | ||
| + | definite context is quite beside the point for mathematics, of course. It is | ||
| + | only in the positive definite case that (anti-) self-dual connections exist and | ||
| + | it is an understanding of the moduli space of these that pays such handsome | ||
| + | topological dividends.<cite>page 377 in Topology, Geometry and Gauge fields by Naber</cite></blockquote> | ||
| - | <note tip> | + | |
| - | The motto in this section is: //the higher the level of abstraction, the better//. | + | [Guid] is Guidry, M., Gauge Field Theories, John Wiley & Sons, Inc., New York, 1991 |
| - | </note> | + | |
| + | |||
| + | <tabbox Examples> | ||
| + | |||
| + | --> Example1# | ||
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| + | |||
| + | <-- | ||
| + | |||
| + | --> Example2:# | ||
| + | |||
| + | |||
| + | <-- | ||
| + | |||
| + | <tabbox FAQ> | ||
| --> Do we really understand Wick rotations?# | --> Do we really understand Wick rotations?# | ||
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| --> Common Question 2# | --> Common Question 2# | ||
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| - | <-- | ||
| - | | ||
| - | <tabbox Examples> | ||
| - | |||
| - | --> Example1# | ||
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| - | <-- | ||
| - | |||
| - | --> Example2:# | ||
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| </tabbox> | </tabbox> | ||
| + | {{tag>theories:quantum_theory:quantum_field_theory}} | ||
advanced_tools/wick_rotation.1497429110.txt.gz · Last modified: 2017/12/04 08:01 (external edit)
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