models:gauge_theory
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models:gauge_theory [2018/04/14 13:40] theodorekorovin [Abstract] |
models:gauge_theory [2018/12/19 11:01] (current) 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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| <tabbox Concrete> | <tabbox Concrete> | ||
| - | [{{ :theories:logicalpatterngauge.png?nolink&300|The logical pattern of a gauge theory. Source: [[http://www.hep.princeton.edu/~mcdonald//examples/EP/mills_ajp_57_493_89.pdf|"Gauge Fields"]] by Robert Mills}}] | + | {{ :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]]. If the gauge symmetry is abelian the Yang-Mills equation reduces to the [[equations:maxwell_equations|Maxwell equations]]. | + | Mathematically Yang-Mills theory is described by the [[equations:yang_mills_equations|Yang-Mills equation]]. |
| + | |||
| + | If the gauge symmetry is abelian the Yang-Mills equation reduces to the [[equations:maxwell_equations|Maxwell equations]]. | ||
| Here is the idea behind gauge theories in a nutshell: | Here is the idea behind gauge theories in a nutshell: | ||
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| 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. | 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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| <tabbox Abstract> | <tabbox Abstract> | ||
| - | {{ :theories:dictionarygauge.png?nolink&600 |Source: [[https://journals.aps.org/prd/abstract/10.1103/PhysRevD.12.3845|Concept of nonintegrable phase factors and global formulation of gauge fields]] by Tai Tsun Wu and Chen Ning Yang}} | ||
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| - | Mathematically, the gauge symmetry means that we a copy of the gauge group above each spacetime point. Each continuous symmetry is at the same time also a manifold. The collection of all these symmetries is called a [[advanced_tools:fiber_bundles|fiber bundle]]. | ||
| - | |||
| - | The gauge fields are [[advanced_tools:connections:ehresmann_connection|Ehresmann connections]] on the bundle, i.e. objects that tell us how we have to move in the bundle when we move through spacetime. Intuitively we can think of the connections as ramps that tells us how the phase of a field changes as it moves through space | ||
| - | |||
| - | [{{ :advanced_tools:fiberramps.png?nolink |[[http://gregnaber.com/wp-content/uploads/GAUGE-FIELDS-AND-GEOMETRY-A-PICTURE-BOOK.pdf|Source]]}}] | ||
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| - | Instead of the the traditional Lagrangian, we can also write the action of a gauge theory using the four-form | ||
| - | \begin{equation} | ||
| - | \mathcal{A}=Tr F\wedge *F=F_{\mu\nu}^aF_{\mu\nu}^a d^4x | ||
| - | \end{equation} | ||
| - | and the action is then | ||
| - | \begin{equation} | ||
| - | \mathcal{S}=\frac {1}{4g^2}\int \mathcal{A} | ||
| - | \end{equation} | ||
| - | |||
| - | ---- | ||
| * 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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| ---- | ---- | ||
| + | **Geometry of Gauge Theories** | ||
| + | |||
| + | |||
| + | |||
| + | [{{ :theories:dictionarygauge.png?nolink&600 |Source: [[https://journals.aps.org/prd/abstract/10.1103/PhysRevD.12.3845|Concept of nonintegrable phase factors and global formulation of gauge fields]] by Tai Tsun Wu and Chen Ning Yang}}] | ||
| + | |||
| + | [{{ :advanced_tools:fiberramps.png?nolink&200|[[http://gregnaber.com/wp-content/uploads/GAUGE-FIELDS-AND-GEOMETRY-A-PICTURE-BOOK.pdf|Source]]}}] | ||
| + | Mathematically, the gauge symmetry means that we a copy of the gauge group above each spacetime point. Each continuous symmetry is at the same time also a manifold. The collection of all these symmetries is called a [[advanced_tools:fiber_bundles|fiber bundle]]. | ||
| + | |||
| + | The gauge fields are [[advanced_tools:connections:ehresmann_connection|Ehresmann connections]] on the bundle, i.e. objects that tell us how we have to move in the bundle when we move through spacetime. Intuitively we can think of the connections as ramps that tells us how the phase of a field changes as it moves through space | ||
| + | |||
| + | The field strength corresponds to the curvature of the fiber bundle. | ||
| + | |||
| + | ---- | ||
| + | |||
| + | **Gauge theory in terms of differential forms** | ||
| + | |||
| + | Instead of the the traditional Lagrangian, we can also write the action of a gauge theory using the four-form | ||
| + | \begin{equation} | ||
| + | \mathcal{A}=Tr F\wedge *F=F_{\mu\nu}^aF_{\mu\nu}^a d^4x | ||
| + | \end{equation} | ||
| + | and the action is then | ||
| + | \begin{equation} | ||
| + | \mathcal{S}=\frac {1}{4g^2}\int \mathcal{A} | ||
| + | \end{equation} | ||
| + | |||
| + | ---- | ||
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| The best theories of nature that we have | The best theories of nature that we have | ||
| - | * [[models:quantum_electrodynamics|Quantum Electrodynamics]] | + | * [[models:standard_model:qed|Quantum Electrodynamics]] |
| - | * [[models:qcd|Quantum Chromodynamics]] | + | * [[models:standard_model:qcd|Quantum Chromodynamics]] |
| * [[models:standard_model|The Standard Model]] | * [[models:standard_model|The Standard Model]] | ||
models/gauge_theory.1523706026.txt.gz · Last modified: 2018/04/14 11:40 (external edit)
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