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advanced_notions:quantum_field_theory:wess-zumino-witten_term [2018/04/13 15:32] ellahughes [Why is it interesting?] |
advanced_notions:quantum_field_theory:wess-zumino-witten_term [2018/04/13 15:37] ellahughes [Intuitive] |
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<blockquote> The W-Z term is a generalization, to the configuration space of scalar fields $\phi_a$, of the | <blockquote> The W-Z term is a generalization, to the configuration space of scalar fields $\phi_a$, of the | ||
charge-monopole interaction term in ordinary configuration space for particles. It acts like a | charge-monopole interaction term in ordinary configuration space for particles. It acts like a | ||
- | monopole in $\phi$-space. | + | monopole in $\phi$-space. [...] |
+ | |||
+ | (a) where the W -Z term itself comes from, or (b) why it is like a monopole in $\phi$-space. The short answer to (a) is: from the very fermion determinant which we studied in the previous lecture, but generalized to $SU(3)_f$, i.e. it is | ||
+ | a term in the effective action for the .P fields which arises after integrating over the fermions [22, 23]. | ||
+ | This is all very well in its way, but it too is mysterious: //why does such an exotic term get induced in | ||
+ | the boson sector when we integrate out the fermions?// The technical answer to this is that the | ||
+ | underlying fermion theory has anomalies, which can be calculated from single fermion loop | ||
+ | diagrams. These diagrams generate effective vertices in the external fields (<i>a, gauge fields, etc.) | ||
+ | coupled to the fermions. Hence any bosonic action obtained by integrating out the fermions- which | ||
+ | is equivalent to summing all single fermion loop diagrams- must faithfully represent these | ||
+ | anomaly-induced vertices. //The W-Z action precisely encodes these anomalous vertices//: if we only | ||
+ | consider the 'ungauged' W -Z action, which is a function of the $SU(3)_f$ chiral field $\phi$ alone, we are | ||
+ | representing correctly just the $SU(3)_f$ flavour anomalies of the underlying Fermion theory. | ||
+ | |||
<cite>Berry Phases, Magnetic Monopoles and {Wess-Zumino} Terms or How the Skyrmion Got Its Spin by I.J.R. Aitchison</cite></blockquote> | <cite>Berry Phases, Magnetic Monopoles and {Wess-Zumino} Terms or How the Skyrmion Got Its Spin by I.J.R. Aitchison</cite></blockquote> | ||
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<blockquote>Witten [8, 9] showed that the Wess-Zumino (W-Z) term [8, 9, 10] in the action for the | <blockquote>Witten [8, 9] showed that the Wess-Zumino (W-Z) term [8, 9, 10] in the action for the | ||
scalar fields $\phi_a$ (whose solitons are [[advanced_notions:skyrmions|Skyrmions]]) actually determines how these solitons are to be | scalar fields $\phi_a$ (whose solitons are [[advanced_notions:skyrmions|Skyrmions]]) actually determines how these solitons are to be | ||
- | quantized. | + | quantized. He obtained the remarkable result that the Skyrmion is a fermion if N, is odd, and a |
+ | boson if N, is even: furthermore, the W-Z term also determines the pattern of spin-SU(3) multiplets $([1/2^+,8],[3/2^+,10],\ldots) in the baryon spectrum [9, 11, 12]. Though obviously correct mathematically, | ||
+ | these results were nevertheless still hard to explain in physical terms, especially to anyone who did not | ||
+ | know what a W -Z term was- and even to those who did") | ||
<cite>Berry Phases, Magnetic Monopoles and {Wess-Zumino} Terms or How the Skyrmion Got Its Spin by I.J.R. Aitchison</cite></blockquote> | <cite>Berry Phases, Magnetic Monopoles and {Wess-Zumino} Terms or How the Skyrmion Got Its Spin by I.J.R. Aitchison</cite></blockquote> |