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advanced_tools:wick_rotation [2017/06/14 10:28]
jakobadmin created
advanced_tools:wick_rotation [2018/03/12 15:29] (current)
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 </​blockquote>​ </​blockquote>​
  
- * [[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. ​+<​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>​ 
 + 
 + 
 +  * [[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. ​
   * To classify all [[http://​jakob.physicsnotes.org/​the_standard_model/​poincare_group#​representations_of_the_lorentz_group|irreducible representations of the Lorentz group]], we must perform a Wick rotation. In this context, the Wick rotation is often called "​Weyl'​s unitary trick"​.   * To classify all [[http://​jakob.physicsnotes.org/​the_standard_model/​poincare_group#​representations_of_the_lorentz_group|irreducible representations of the Lorentz group]], we must perform a Wick rotation. In this context, the Wick rotation is often called "​Weyl'​s unitary trick"​.
   * Path integrals only make sense after we perform a Wick rotation:   * Path integrals only make sense after we perform 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>​ +
-  +
-<tabbox Researcher> ​+
  
-<note tip> +<cite>https://arxiv.org/pdf/1311.0813.pdf</​cite>​ 
-The motto in this section is: //the higher the level of abstraction,​ the better//. +</blockquote>
-</note>+
  
---> Common Question 1# 
  
-  +--> Wick Rotation in Classical Mechanics#
-<--+
  
---Common Question 2#+<​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> ​
 +<​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>​
 +
 +
 +[Guid] is Guidry, M., Gauge Field Theories, John Wiley & Sons, Inc., New York, 1991
 +
   ​   ​
 <tabbox Examples> ​ <tabbox Examples> ​
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 --> Example2:# --> Example2:#
 +
 + 
 +<--
 +
 +<tabbox FAQ> ​
 +
 +--> Do we really understand Wick rotations?#
 +
 +
 +
 +Although a Wick Rotation is a standard tool in QFT not all aspects seem to be sufficiently understood: ​
 +
 +<​blockquote>​
 +Another peculiarity of chiral theories arises when one tries to understand how they behave under Wick rotation. Non-perturbative QFT calculations are well-defined not in Minkowski space, but in Euclidean space, with physical observables recovered by analytic continuation. But the behavior of spinors in Minkowski and Euclidean space is quite different, leading to a very confusing situation. Despite several attempts over the years to sort this out for myself, I remain confused, and can’t help suspecting that there is more to this than a purely technical problem. One natural mathematical setting for trying to think about this is the twistor formalism, where complexified,​ compactified Minkowski space is the Grassmanian of complex 2-planes in complex 4-space. The problem though is that thinking this way requires taking as basic variables holomorphic quantities, and how this fits into the standard QFT formalism is unclear. Perhaps the current vogue for twistor methods to study gauge-theory amplitudes will shed some light on this. On the general problem of Wick rotation, about the deepest thinking that I’ve seen has been that of Graeme Segal, who deals with the issue in the 2d context in his famous manuscript “The Definition of Conformal Field Theory”.
 +
 +<​cite>​https://​www.math.columbia.edu/​~woit/​wordpress/?​p=2876</​cite>​
 +</​blockquote>​
 +
 +<​blockquote>​
 +I’ve always thought this whole confusion is an important clue that there is something about the relation of QFT and geometry that we don’t understand. Things are even more confusing than just worrying about Minkowski vs. Euclidean metrics. To define spinors, we need not just a metric, but a spin connection. In Minkowski space this is a connection on a Spin(3,​1)=SL(2,​C) bundle, in Euclidean space on a Spin(4)=SU(2)xSU(2) bundle, and these are quite different things, with associated spinor fields with quite different properties. **So the whole “Wick Rotation” question is very confusing even in flat space-time when one is dealing with spinors.**
 +
 +<​cite>​https://​www.math.columbia.edu/​~woit/​wordpress/​archives/​000160.html</​cite>​
 +</​blockquote>​
 +
 + 
 +<--
 +
 +--> Common Question 2#
  
    
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 </​tabbox>​ </​tabbox>​
  
 +{{tag>​theories:​quantum_theory:​quantum_field_theory}}
  
advanced_tools/wick_rotation.1497428935.txt.gz · Last modified: 2017/12/04 08:01 (external edit)