advanced_notions:quantum_field_theory:wess-zumino-witten_term
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advanced_notions:quantum_field_theory:wess-zumino-witten_term [2018/04/13 15:29] ellahughes [Concrete] |
advanced_notions:quantum_field_theory:wess-zumino-witten_term [2018/04/13 15:58] (current) ellahughes [Intuitive] |
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| <tabbox Intuitive> | <tabbox Intuitive> | ||
| - | <note tip> | + | <blockquote> The W-Z term is a generalization, to the configuration space of scalar fields $\phi_a$, of the |
| - | Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party. | + | charge-monopole interaction term in ordinary configuration space for particles. It acts like a |
| - | </note> | + | 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. | ||
| + | |||
| + | [...] | ||
| + | |||
| + | our concern here has been to place the 'monopole' form (5 .15) of $L_{W-Z}$ in the context of an adiabatic | ||
| + | decoupling problem. From this point of view, the peculiar phase behaviour leading to 'fermion-ness' | ||
| + | in the $\phi$, sector has arisen as a result of non-trivial structure left behind when the fermion vacuum is | ||
| + | decoupled adiabatically from the $\phi$'s. If we use only the $\phi$ d.f. 's, and integrate the fermions away, we | ||
| + | must include a W-Z term which embodies this structure. The ultimate reason that this structure has a | ||
| + | 'monopole' form is to be found in the topological approach to anomalies [27, 28]. | ||
| + | |||
| + | <cite>Berry Phases, Magnetic Monopoles and {Wess-Zumino} Terms or How the Skyrmion Got Its Spin by I.J.R. Aitchison</cite></blockquote> | ||
| | | ||
| <tabbox Concrete> | <tabbox Concrete> | ||
| - | <blockquote>Properties of WZ terms< | + | <blockquote>**Properties of WZ terms** |
| * (i)are metric independent | * (i)are metric independent | ||
| Line 16: | Line 39: | ||
| * (iv) do not depend on m – the scale, below which an effective action is valid (but do depend on sgn (m)) | * (iv) do not depend on m – the scale, below which an effective action is valid (but do depend on sgn (m)) | ||
| * (v) are antisymmetric in derivatives with respect to different space-time coordinates (contain !µνλ...) | * (v) are antisymmetric in derivatives with respect to different space-time coordinates (contain !µνλ...) | ||
| - | * (vi) are written as integrals of (D+1)-forms over auxiliary (D+1)-dimensional space - disk $^{DD+1}$ such that ∂$^{DD+1}$ = $S^D$ - compactified space-time | + | * (vi) are written as integrals of (D+1)-forms over auxiliary (D+1)-dimensional space - disk $D^{D+1}$ such that ∂$D^{D+1}$ = $S^D$ - compactified space-time |
| * (vii) are multi-valued functionals. Multi-valuedness results in quantization of coupling constants (coefficients in front of WZ terms) | * (vii) are multi-valued functionals. Multi-valuedness results in quantization of coupling constants (coefficients in front of WZ terms) | ||
| * (viii) do change equations of motion by changing commutation relation between fields (Poisson’s brackets) not by changing Hamiltonian | * (viii) do change equations of motion by changing commutation relation between fields (Poisson’s brackets) not by changing Hamiltonian | ||
| Line 24: | Line 47: | ||
| * (xii) can be calculated by gradient expansion of the variation of fermionic determinants | * (xii) can be calculated by gradient expansion of the variation of fermionic determinants | ||
| * (xiii) produce θ terms as a reduction of target space | * (xiii) produce θ terms as a reduction of target space | ||
| + | |||
| + | Among the listed properties the first five (i)-(v) are the properties of all | ||
| + | topological terms while the others are more specific to WZ terms. | ||
| <cite>[[http://felix.physics.sunysb.edu/~abanov/Teaching/Spring2009/Notes/abanov-cp07-upload.pdf|WZW term in quantum mechanics: single spin]]</cite></blockquote> | <cite>[[http://felix.physics.sunysb.edu/~abanov/Teaching/Spring2009/Notes/abanov-cp07-upload.pdf|WZW term in quantum mechanics: single spin]]</cite></blockquote> | ||
| Line 33: | Line 59: | ||
| </note> | </note> | ||
| - | <tabbox Why is it interesting?> | + | <tabbox Why is it interesting?> |
| + | | ||
| <blockquote> | <blockquote> | ||
| - | The chiral anomaly can be corrected by adding a Wess-Zumino term to the Lagrangian, but this term is not perturbatively renormalizable, thus does not solve the nonrenormalizability problem. | + | The chiral [[advanced_notions:quantum_field_theory:anomalies|anomaly]] can be corrected by adding a Wess-Zumino term to the Lagrangian, but this term is not perturbatively renormalizable, thus does not solve the nonrenormalizability problem. |
| <cite>https://physics.stackexchange.com/a/34022/37286</cite> | <cite>https://physics.stackexchange.com/a/34022/37286</cite> | ||
| </blockquote> | </blockquote> | ||
| + | |||
| + | <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 | ||
| + | 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> | ||
| </tabbox> | </tabbox> | ||
advanced_notions/quantum_field_theory/wess-zumino-witten_term.1523626163.txt.gz · Last modified: 2018/04/13 13:29 (external edit)
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