advanced_tools:wick_rotation
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advanced_tools:wick_rotation [2017/11/14 15:50] jakobadmin [Researcher] |
advanced_tools:wick_rotation [2018/03/12 15:29] (current) jakobadmin |
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| + | [Guid] is Guidry, M., Gauge Field Theories, John Wiley & Sons, Inc., New York, 1991 | ||
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| + | <-- | ||
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| + | <tabbox FAQ> | ||
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| + | --> Do we really understand Wick rotations?# | ||
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| + | Although a Wick Rotation is a standard tool in QFT not all aspects seem to be sufficiently understood: | ||
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| + | <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”. | ||
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| + | <cite>https://www.math.columbia.edu/~woit/wordpress/?p=2876</cite> | ||
| + | </blockquote> | ||
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| + | <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.** | ||
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| + | <cite>https://www.math.columbia.edu/~woit/wordpress/archives/000160.html</cite> | ||
| + | </blockquote> | ||
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| + | <-- | ||
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| + | --> Common Question 2# | ||
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| + | {{tag>theories:quantum_theory:quantum_field_theory}} | ||
advanced_tools/wick_rotation.1510671055.txt.gz · Last modified: 2017/12/04 08:01 (external edit)
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