QFT review sheet

Basic Ideas

  1. What are fields? (diff between theory of field and theory of particles)

  2. Why quantum field theory?

    1. Desire (photon and phonon)

    2. Marriage of QM and R. (# of part. can change)

    3. Fields maybe more fundemantal.

    4. Connection between QFT and classical thermal field theory at \(T>0\).

  3. How to quantize? discretize, solve N-body, take limit.

  4. \(\mathbf{x}_i(t) \Longleftrightarrow \phi(\mathbf{x}, t)\)

Klein-Gordon Equation

A relativistic one particle Shrödinger Equation: (Also describes acoustic phonons.) \[\hbar^2 (-\partial_{ct}^2 +\nabla^2) \psi = M^2 c^2 \psi\] \[-\hbar^2\partial^\mu\partial_\mu \psi = M^2c^2 \psi\]

Problems with K-G.

Problem: \(|\psi|^2\) is no longer prob. density. Not conserved.

What is conserved? \(\rho = i(\psi^* \dot{\psi} - \dot{\psi}^*\psi)\). However, this can be negative.

Also, energy have negative solutions.

Classical limit: \(\ddot{\phi} = (c^2\nabla^2 -\frac{(Mc^2)^2}{\hbar^2})\phi \Longrightarrow \ddot{\phi} = (c^2\nabla^2 - \omega_{min}^2)\phi\)

K-G arises in CM physics

  1. \(M=0\) case: \(\ddot{\phi} = c^2 \nabla^2 \phi\). Dispersion relation: (Note, \(c\) is the wave speed, not speed of light.) \[\omega^2 = c^2 k^2\] Arbitratily low frequency waves \(\Longrightarrow\) “acoustic limit”

  2. \(M\ne 0\) case: \(\ddot{\phi} = (c^2\nabla^2 - \omega_{min}^2)\phi\). Dispersion relation: (“optical” branches) \[\omega^2 = c^2 k^2 + \omega_{min}^2\]

  3. Nomenclature: gap.

Quantizing K-G equation: Basic concepts

\(\ddot{\phi} = (\nabla^2 - m^2)\phi\)

  1. Action: (Lagrangian density and Lagrangian) \[S[\phi(\mathbf{x},t)] = \int_{t_1}^{t_2} dt \int d^3 x \mathcal{L}(\phi(\mathbf{x},t), \dot{\phi}(\mathbf{x},t), \nabla\phi(\mathbf{x},t))\]


    1. Small transverse fluctuations of a violin strings. \(\mathcal{L} = \frac{1}{2}\rho \dot{\phi}^2 - \frac{1}{2} T (\phi')^2\)

    2. EM in \(A_0 = 0\) gauge. \(\mathcal{L} = -\frac{1}{4}F_{\mu \nu}F^{\mu \nu}\)

  2. Eular-Lagrangian Equation. Excise: Derive this from Hamilton’s principle, or variation principle. Cf. Page 7.

    \[\partial_t \frac{\partial \mathcal{L}}{\partial (\partial_t \phi)} + \nabla\cdot \frac{\partial \mathcal{L}}{\partial (\nabla \phi)} = \frac{\partial \mathcal{L}}{\partial \phi}\]

    Relativistic version:

    \[\partial_\mu \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} = \frac{\partial \mathcal{L}}{\partial \phi}\]

    Example on \(\partial_\mu\) and on a given Lagrangian.

    Note: different people treat different terms as K.E. or P.E..

  3. Lagrangian – Hamiltonian

    Review: \(H = \sum_i p_i \dot{q}_i - L\). Quantization: \([\hat{p}_i, \hat{q}_j] = -i\hbar \delta_{ij}\)

    Canonical “momentum” field conjugate to \(\phi(\mathbf{x})\): \[\pi(\mathbf{x})\equiv \frac{\partial \mathcal{L}}{\partial \dot{\phi}}\]

    Moral: canonical momentum fields are often not related to what you think of as actual momentum.

    \[H = \left(\int d^3 x \pi(\mathbf{x}, t)\dot{\phi}(\mathbf{x}, t)\right) - L = \int d^3 x \mathcal{H}\] with \[\mathcal{H} = \pi(\mathbf{x}, t)\dot{\phi}(\mathbf{x}, t) - \mathcal{L}\]

    Quantization: (the rest are \(0\)) \[[\hat{\pi}(\mathbf{x}), \hat{\phi}(\mathbf{y})] = - i \hbar \delta^{(3)} (\mathbf{x} - \mathbf{y})\]

    Example: For \(\mathcal{L} = \frac{1}{2}\dot{\phi}^2 - \frac{1}{2}(\nabla \phi)^2 - \frac{1}{2}m^2 \phi^2\): \[\mathcal{H} = \frac{1}{2} \pi^2 + \frac{1}{2} (\nabla \phi)^2 + \frac{1}{2} m^2 \phi^2\] Note: this is not Lorentz inv.

Quantize Klein-Gordon

Fourier Transformation

\[\hat{\phi}(\mathbf{x}) = \int \frac{d^3 k}{(2\pi)^3 } \hat{\tilde{\phi}}(\mathbf{k})e^{i\mathbf{k}\cdot \mathbf{x}}\] \[\hat{\pi}(\mathbf{y}) = \int \frac{d^3 k}{(2\pi)^3 } \hat{\tilde{\pi}}(\mathbf{k})e^{i\mathbf{k}\cdot \mathbf{y}}\]

So (excise: Prove this): \[\hat{H} = \int \frac{d^3 k}{(2\pi)^3 } \left[ \frac{1}{2}\hat{\tilde{\pi}}(\mathbf{k})^\dagger \hat{\tilde{\pi}}(\mathbf{k}) +\frac{1}{2}\omega_\mathbf{k}^2 \hat{\tilde{\phi}}(\mathbf{k})^\dagger \hat{\tilde{\phi}}(\mathbf{k}) \right]\]

Note: These are complex harmonic oscillators. \(\tilde{\phi}(-\mathbf{k})=\tilde{\phi}(\mathbf{k})^*\). All \(\mathbf{k}\)’s are not independent.

Commutation Relations

\[[\hat{\tilde{\pi}}(\mathbf{k'}), \hat{\tilde{\phi}}(\mathbf{k})] = - i (2\pi)^3 \delta^{(3)} (\mathbf{k} + \mathbf{k'})\] Or: \[[\hat{\tilde{\pi}}(\mathbf{k'})^\dagger, \hat{\tilde{\phi}}(\mathbf{k})] = - i (2\pi)^3 \delta^{(3)} (\mathbf{k} - \mathbf{k'})\]

Defining uppering and lowering operators, we get: \[\hat{\phi}(\mathbf{x}) = \int \frac{d^3 k}{(2\pi)^3 } \frac{1}{\sqrt{2\omega_\mathbf{k}}}\left[ \hat{a}_\mathbf{k} e^{i\mathbf{k}\cdot\mathbf{x}} + \hat{a}_\mathbf{k}^\dagger e^{-i\mathbf{k}\cdot \mathbf{x}} \right]\] Note the sign difference, as \(\hat{\phi}(\mathbf{x})\) must be Hermitian.

\[[a_\mathbf{k}, a_\mathbf{k'}^\dagger] = (2\pi)^3 \delta(\mathbf{k} - \mathbf{k'})\] \[\hat{H} = \int \frac{d^3 k}{(2\pi)^3 } \omega_\mathbf{k} \left(a_\mathbf{k}^\dagger a_\mathbf{k'} + \frac{1}{2}\right)\]

Ex: check this. \(a_\mathbf{k}\) and \(a_\mathbf{k}^\dagger\) are also called annihilation and creation operators.

Note: the 1/2 in the above expression is actually \((1/2)\delta^{(3)}(0)\). There is some argument on discretized vs continuous normalization concerning this topic.

Aside: regularization and renormalization. Find tuning problem. Dirac sea. Supersymmetry.

Noether Theorem and Conservation

Starting point: \(a_\mathbf{k}^\dagger\) should also increase the momentum of the system by \(\hbar \mathrm{k}\).

Transformation and Symmetry

  1. Lorentz transform: \[x^\mu \rightarrow \Lambda^\mu_{\,\,\nu}x^\nu\]

    It can be a boost or a rotation or a combination of both. Inverse: \(\Lambda^{-1}\). (\(\beta \to -\beta\) for example)

  2. All Lorentz Transformations satisfy: \[\eta \Lambda \eta = (\Lambda^{-1})^T \qquad \mathrm{or} \qquad \eta\Lambda = (\Lambda^{-1})^T \eta\]

  3. Invariant thing: \(X^T \eta Y\) (Ex: Prove this)

  4. Notation of covariance and contravariance. \(\eta^\mu_{\,\,\nu} \equiv \delta^\mu_\nu\). \[\eta x \to (\Lambda^{-1})^T\eta x\]