advanced_tools:wick_rotation
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advanced_tools:wick_rotation [2017/06/14 10:29] jakobadmin [Why is it interesting?] |
advanced_tools:wick_rotation [2018/03/12 15:29] (current) jakobadmin |
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| <cite>http://jfi.uchicago.edu/~leop/TALKS/Phase%20TransitionsV2.4Dirac.pdf</cite> | <cite>http://jfi.uchicago.edu/~leop/TALKS/Phase%20TransitionsV2.4Dirac.pdf</cite> | ||
| </blockquote> | </blockquote> | ||
| + | |||
| + | <blockquote> | ||
| + | This procedure (Wick rotation), which relates classical statistical mechanics and quantum field theory, is, however, somewhat subtle | ||
| + | |||
| + | <cite>https://arxiv.org/pdf/hep-th/9802035.pdf</cite> | ||
| + | </blockquote> | ||
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| * [[http://jakob.physicsnotes.org/quantum_field_theory/methods/non_perturbative_qft#why_imaginary_time|Tunneling phenomena]] are described most easily by performing a Wick rotation. | * [[http://jakob.physicsnotes.org/quantum_field_theory/methods/non_perturbative_qft#why_imaginary_time|Tunneling phenomena]] are described most easily by performing a Wick rotation. | ||
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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> | ||
| + | |||
| + | |||
| + | --> Wick Rotation in Classical Mechanics# | ||
| + | |||
| + | <blockquote> | ||
| + | One of the stranger aspects of Lagrangian dynamics is how it turns into statics when we replace the time coordinate t by it — or in the jargon of physicists, when we ‘Wick rotate’ to ‘imaginary time’! People usually take advantage of this to do interesting things in the context of quantum mechanics, but the basic ideas are already visible in classical mechanics. | ||
| + | |||
| + | <cite>http://math.ucr.edu/home/baez/classical/spring.pdf</cite> | ||
| + | </blockquote> | ||
| + | |||
| + | <-- | ||
| <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# | ||
| + | |||
| + | |||
| + | <-- | ||
| + | |||
| + | --> 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# | ||
| - | |||
| - | |||
| - | <-- | ||
| - | |||
| - | --> Example2:# | ||
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| </tabbox> | </tabbox> | ||
| + | {{tag>theories:quantum_theory:quantum_field_theory}} | ||
advanced_tools/wick_rotation.1497428986.txt.gz · Last modified: 2017/12/04 08:01 (external edit)
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