Quantum Field Theory | ||
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... no spooky action at a distance (Einstein) | ||
Early Results
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Relativistic Quantum Mechanics | Klein-Gordon Equation Dirac Equation | |
The Dawn of QFT | Spinors Spin Feynman Slash Notation Antimatter Klein-Gordon Field Dirac Field Renormalisation Grassman Variable Conformal Field Theory | |
Countdown to the Standard Model
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From a framework to a model | Yang-Mills Theory Quantum Electrodynamics Quantum Chromodynamics Electroweak Theory Higgs Mechanism Standard Model | |
Semi-Classical Gravity and the Dark Age | Hawking Radiation Chandrashekhar Limit Inflation Problems with the Standard Model | |
Outlook
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Beyond the Standard Model | Beyond the Standard Model Quantum Gravity Theory of Everything | |
Related
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Related | De Donder-Weyl Theory | |
Quantum Electrodynamics, or QED in short, is a Quantum Field Theory relating to Electromagnetism and Light. It is a weakly-coupled theory, with a Fine-Structure Constant of $ \frac{1}{137} $ as it's Coupling Constant yet there is a need for Renormalisation. Quantum Electrodynamics has a gauge group of $ U(1) $.
Lagrangian DensityEdit this section
As is seen with Quantum Chromodynamics (QCD), too, the Lagrangian Density for Quantum Electrodynamics for a spin-1/2 field interacting with the Electromagnetic Field is:
Lagrangian Density of Quantum Electrodynamics - $ \mathcal{L}=\bar\psi(i\not D -m)\psi -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} $
Field EquationEdit this section
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To begin, substituting the definition of $ D $ into the Lagrangian gives us
- $ \mathcal{L} = i \bar\psi \gamma^\mu \partial_\mu \psi - e\bar{\psi}\gamma_\mu (A^\mu+B^\mu) \psi -m \bar{\psi} \psi - \frac{1}{4}F_{\mu\nu}F^{\mu\nu}. \, $
Next, we can substitute this Lagrangian into the Euler–Lagrange Equation of motion for a field:
- $ \partial_\mu \left( \frac{\partial \mathcal{L}}{\partial ( \partial_\mu \psi )} \right) - \frac{\partial \mathcal{L}}{\partial \psi} = 0 \, $ (2)
to find the field equations for QED.
The two terms from this Lagrangian are then
- $ \partial_\mu \left( \frac{\partial \mathcal{L}}{\partial ( \partial_\mu \psi )} \right) = \partial_\mu \left( i \bar{\psi} \gamma^\mu \right), \, $
- $ \frac{\partial \mathcal{L}}{\partial \psi} = -e\bar{\psi}\gamma_\mu (A^\mu+B^\mu) - m \bar{\psi}. \, $
Substituting these two back into the Euler–Lagrange equation (2) results in
- $ i \partial_\mu \bar{\psi} \gamma^\mu + e\bar{\psi}\gamma_\mu (A^\mu+B^\mu) + m \bar{\psi} = 0 \, $
with complex conjugate
- $ i \gamma^\mu \partial_\mu \psi - e \gamma_\mu (A^\mu+B^\mu) \psi - m \psi = 0. \, $
Bringing the middle term to the right-hand side transforms this second equation into
$ i \gamma^\mu \partial_\mu \psi - m \psi = e \gamma_\mu (A^\mu+B^\mu) \psi \, $
The left-hand side is like the original Dirac Equation and the right-hand side is the interaction with the electromagnetic field.
One further important equation can be found by substituting the Lagrangian into another Euler–Lagrange equation, this time for the field, $ A^\mu $:
$ \partial_\nu \left( \frac{\partial \mathcal{L}}{\partial ( \partial_\nu A_\mu )} \right) - \frac{\partial \mathcal{L}}{\partial A_\mu} = 0\,. $(3)
The two terms this time are
- $ \partial_\nu \left( \frac{\partial \mathcal{L}}{\partial ( \partial_\nu A_\mu )} \right) = \partial_\nu \left( \partial^\mu A^\nu - \partial^\nu A^\mu \right), \, $
- $ \frac{\partial \mathcal{L}}{\partial A_\mu} = -e\bar{\psi} \gamma^\mu \psi \, $
and these two terms, when substituted back into (3) give us
$ \partial_\nu F^{\nu \mu} = e \bar{\psi} \gamma^\mu \psi \, $
Now, if we impose the Lorenz Gauge Condition, that the divergence of the four potential vanishes
- $ \partial_{\mu} A^\mu = 0 $
then we get
- $ \Box A^{\mu}=e\bar{\psi} \gamma^{\mu} \psi\,, $
which is a Wave Equation for the four potential, the Quantum Electrodynamical version of the classical Maxwell equations in the Lorenz gauge.
RenormalisationEdit this section
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Interaction PictureEdit this section
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This theory can be straightforwardly quantized by treating bosonic and fermionic sectors as free. This permits us to build a set of asymptotic states which can be used to start a computation of the probability amplitudes for different processes. In order to do so, we have to compute a time-evolution operator that, for a given initial state $ |i\rangle $, will give a final state $ \langle f| $ in such a way to have
- $ M_{fi}=\langle f|U|i\rangle. $
This technique is also known as the S-Matrix. The evolution operator is obtained in the Interaction Picture where time evolution is given by the interaction Hamiltonian, which is the integral over space of the second term in the Lagrangian density given above:
- $ V=e\int d^3x\bar\psi\gamma^\mu\psi A_\mu $
and so, one has
- $ U=T\exp\left[-\frac{i}{\hbar}\int_{t_0}^tdt'V(t')\right] $
where $ T $ is the time ordering operator. This evolution operator only has meaning as a series, and what we get here is a Perturbation Series with the Fine-Structure Constant as the development parameter. This series is called the Dyson Series.