models:gauge_theory
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models:gauge_theory [2018/05/06 11:47] jakobadmin [Concrete] |
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| - | ====== Gauge Theory ====== | + | ====== Gauge Models====== |
| - | //also known as Yang-Mills theory// | + | //also known as Yang-Mills Models// |
| <tabbox Intuitive> | <tabbox Intuitive> | ||
| - | A gauge theory is a theory where the way different particles or fields interact with each other is determined by a [[advanced_tools:gauge_symmetry|gauge symmetry]]. | + | A gauge model is a model where the way different particles or fields interact with each other is determined by a [[advanced_tools:gauge_symmetry|gauge symmetry]]. |
| The particles that mediate the gauge interaction are called gauge bosons. | The particles that mediate the gauge interaction are called gauge bosons. | ||
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| {{ :theories:gaugetheornoether.png?nolink&300|}} | {{ :theories:gaugetheornoether.png?nolink&300|}} | ||
| - | The basic idea behind gauge theories is that the fundamental symmetries of nature actually dictate the interactions between the fields of nature. | + | The basic idea behind gauge models is that the fundamental symmetries of nature actually dictate the interactions between the fields of nature. |
| Mathematically Yang-Mills theory is described by the [[equations:yang_mills_equations|Yang-Mills equation]]. | Mathematically Yang-Mills theory is described by the [[equations:yang_mills_equations|Yang-Mills equation]]. | ||
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| Consider now ‘localizing’ these phase transformations, i.e. letting Λ become an arbitrary function of the coordinates \Lambda(x): \Psi \to e^{iq\Lambda(x)} \Psi, \bar \Psi \to e^{-iq\Lambda(x)} \bar \Psi. As it stands, the free field Lagrangian is clearly not invariant under such transformations, since the derivatives of the arbitrary functions, i.e. ∂μ \Lambda(x), will not vanish in general. The Lagrangian must be modified if the theory is to admit the local transformations as (variational) symmetries. In particular, we replace the free field Lagrangian with L_{interacting} = \bar \Psi(i\gamma^\mu ∂μ − m)\Psi − q A_μ \bar \Psi γ^\mu \Psi ≡ L_{Dirac} − J_μ A^μ , with J_μ = q \bar \Psi \gamma_\mu \Psi. This current is in fact the conserved current associated with the global U(1) invariance of the interacting theory. Towards securing local invariance we have introduced the field A_μ, the gauge potential. The particular form of coupling between the matter field and this gauge potential in L_{interacting} is termed minimal coupling. | Consider now ‘localizing’ these phase transformations, i.e. letting \Lambda become an arbitrary function of the coordinates \Lambda(x): \Psi \to e^{iq\Lambda(x)} \Psi, \bar \Psi \to e^{-iq\Lambda(x)} \bar \Psi. As it stands, the free field Lagrangian is clearly not invariant under such transformations, since the derivatives of the arbitrary functions, i.e. ∂μ \Lambda(x), will not vanish in general. The Lagrangian must be modified if the theory is to admit the local transformations as (variational) symmetries. In particular, we replace the free field Lagrangian with L_{interacting} = \bar \Psi(i\gamma^\mu ∂μ − m)\Psi − q A_μ \bar \Psi γ^\mu \Psi ≡ L_{Dirac} − J_μ A^μ , with J_μ = q \bar \Psi \gamma_\mu \Psi. This current is in fact the conserved current associated with the global U(1) invariance of the interacting theory. Towards securing local invariance we have introduced the field A_μ, the gauge potential. The particular form of coupling between the matter field and this gauge potential in L_{interacting} is termed minimal coupling. | ||
| - | This modified Lagrangian is now invariant under the local phase transformations provided that the vector field A_μ is simultaneously transformed according to A_μ (x) → A_μ(x) − ∂μ \Lambda(x). | + | This modified Lagrangian is now invariant under the local phase transformations provided that the vector field A_μ is simultaneously transformed according to $A_μ (x) → A_μ(x) − ∂_μ \Lambda(x)$. |
| electromagnetic potential. | electromagnetic potential. | ||
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| * A great introduction to the geometric perspective on gauge theories can be found in “Geometrical methods of mathematical physics” by Bernard F. Schutz at page 219 | * A great introduction to the geometric perspective on gauge theories can be found in “Geometrical methods of mathematical physics” by Bernard F. Schutz at page 219 | ||
| + | * See also [[https://www.claymath.org/sites/default/files/yangmills.pdf|Quantum Yang-Mills Theory by Jaffe and Witten]] | ||
| * Another great introduction to the **fibre bundle formulation** of gauge theories with many nice pictures is [[https://arxiv.org/pdf/1607.03089.pdf|Gauge Theories and Fiber Bundles: Definitions, Pictures, and Results]] by Adam Marsh | * Another great introduction to the **fibre bundle formulation** of gauge theories with many nice pictures is [[https://arxiv.org/pdf/1607.03089.pdf|Gauge Theories and Fiber Bundles: Definitions, Pictures, and Results]] by Adam Marsh | ||
| * A pedagogical introduction to the **loop formulation** of gauge theory can be found in the book Some Elementary Gauge Theory Concepts by Sheung Tsun Tsou, Hong-Mo Chan. | * A pedagogical introduction to the **loop formulation** of gauge theory can be found in the book Some Elementary Gauge Theory Concepts by Sheung Tsun Tsou, Hong-Mo Chan. | ||
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| * Baez, J. and Muniain, J. P.: Gauge Fields, Knots and Gravity. World Scientific Publishing Co Inc, 1994. and especially, the [[http://michaelnielsen.org/blog/introduction-to-yang-mills-theories/|nice summary by Michael Nielsen]]. | * Baez, J. and Muniain, J. P.: Gauge Fields, Knots and Gravity. World Scientific Publishing Co Inc, 1994. and especially, the [[http://michaelnielsen.org/blog/introduction-to-yang-mills-theories/|nice summary by Michael Nielsen]]. | ||
| * http://www.damtp.cam.ac.uk/user/dbs26/AQFT/YM.pdf | * http://www.damtp.cam.ac.uk/user/dbs26/AQFT/YM.pdf | ||
| + | * [[https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.52.175|The geometrical setting of gauge theories of the Yang-Mills]] type by M. Daniel and C. M. Viallet | ||
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