Noether's Theorem ================================ Noether's theorem deals with continuous symmetry. Noether's Theorem of Particles ---------------------------------------------------- Noether's Theorem of Fields ---------------------------------------------------- Suppose we have a continuous transformation, which is internal, that transforms the fields according to .. math:: \phi_i (x^\mu) \to \phi_i (x^\mu) + \delta \phi_i(x^\mu). For convenience, we explicity write the variation :math:`\delta \phi_i(x^\mu)` as a continuous quantity :math:`\alpha`, i.e., .. math:: \delta \phi_i(x^\mu) = \alpha \Delta\phi(x^\mu). Noether's theorem states that if this continuous preserves the Lagrangian, we can define conserved Noether current thus conserved charge. .. admonition:: Planning the Proof :class: toggle 1. Write down the variation of Lagrangian. 2. Combine the terms and apply the Euler-Lagrangian equation. 3. If the Lagrangian is invariant under such a continuous tranformation, blablabla. The variation of Lagrangian is .. math:: \delta \mathcal L(\phi_i, \dot \phi_i) = \frac{\partial \mathcal L}{\partial \phi_i} \delta \phi_i + \frac{\partial \mathcal L}{\partial \phi_{i,\mu}} \delta(\phi_{i,\mu}). :label: variation-of-lagrangian-fields We know that the variation and partial derivative can be exchanged, such that .. math:: \delta(\phi_{i,\mu}) = \partial_\mu (\delta \phi_i). We rewrite the second term in Eq. |nbsp| :eq:`variation-of-lagrangian-fields`, .. math:: &\frac{\partial \mathcal L}{\partial \phi_{i,\mu}} \delta(\phi_{i,\mu}) \\ = & \frac{\partial \mathcal L}{\partial \phi_{i,\mu}} \partial_\mu( \delta \phi_{i} ) \\ = & \partial_\mu\left( \frac{\partial \mathcal L}{\partial \phi_{i,\mu}} \delta \phi_{i} \right) - \delta \phi_{i} \partial_\mu \left( \frac{\partial \mathcal L}{\partial \phi_{i,\mu}} \right). Plug it back into Eq. |nbsp| :eq:`variation-of-lagrangian-fields`, we have .. math:: \delta \mathcal L(\phi_i, \dot \phi_i) =& \frac{\partial \mathcal L}{\partial \phi_i} \delta \phi_i + \partial_\mu\left( \frac{\partial \mathcal L}{\partial \phi_{i,\mu}} \delta \phi_{i} \right) - \delta \phi_{i} \partial_\mu \left( \frac{\partial \mathcal L}{\partial \phi_{i,\mu}} \right) \\ = & \delta \phi_i \left( \frac{\partial \mathcal L}{\partial \phi_i} - \partial_\mu \left( \frac{\partial \mathcal L}{\partial \phi_{i,\mu}} \right) \right) + \partial_\mu\left( \frac{\partial \mathcal L}{\partial \phi_{i,\mu}} \delta \phi_{i} \right). The first term is zero by Euler-Lagrangian equation. Thus .. math:: \delta \mathcal L(\phi_i, \dot \phi_i) =\partial_\mu\left( \frac{\partial \mathcal L}{\partial \phi_{i,\mu}} \delta \phi_{i} \right). Now we impose the condition that the Lagrangian is invariant under such a continuous transformation, so that :math:`\delta \mathcal L = 0`. .. math:: \partial_\mu\left( \frac{\partial \mathcal L}{\partial \phi_{i,\mu}} \delta \phi_{i} \right) = 0. :label: constant-of-motion-fields-1 Eq. |nbsp| :eq:`constant-of-motion-fields-1` defines the :highlight-text:`constant of motion`. Put the definition of :math:`\delta \phi_i` back in, .. math:: \alpha \partial_\mu\left( \frac{\partial \mathcal L}{\partial \phi_{i,\mu}} \Delta \phi_{i} \right) = 0. :label: constant-of-motion-fields Examples of Noether Current ---------------------------------------------- Global Phase Transformation ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ For the Lagrangian .. math:: \mathcal L = \partial^\mu \phi^* \partial_\mu \phi - m^2 \phi^* \phi, that leads to Klein-Gordon equation, a transformation .. math:: \phi \to e^{i\alpha}\phi,\\ \phi^* \to e^{-i\alpha}\phi^*, will not change the scalar particle Lagrangian. The corresponding Noether current is defined by .. math:: \partial_\mu j^\mu = 0, where .. math:: j^\mu = -i(\phi^* \partial^\mu \phi - \phi\partial^\mu \phi^*). Along with the current we find the conserved charge .. math:: Q = \int d^3 x j^0, which satisfies .. math:: \frac{\partial Q}{\partial t}= 0. .. admonition:: Proof :class: toggle Here is the proof. Space-time Translation ~~~~~~~~~~~~~~~~~~~~~~~~ For arbitary Lagrangian :math:`\mathcal L(x^\mu)` which is space-time dependent, we can calculate the action .. math:: S = \int d^4x \mathcal L. If the action is invariant under space-time translation .. math:: x^\mu\to x^\mu + \alpha a^\mu, we find the conserved current to be the energy-momentum tensor :math:`T^{\mu\nu}` .. math:: T^{\mu\nu} = \frac{\partial \mathcal L}{\partial (\partial_\mu\phi)}\partial^\nu \phi - g^{\mu\nu} \mathcal L. The corresponding conservation equation is .. math:: \partial_\mu T^{\mu\nu} = 0, which defines the four charges .. math:: Q^\mu = \int d^3 T^{\mu\nu} . .. admonition:: Proof Energy-momentum Tensor as Noether Current :class: toggle QED. For the Lagrangian .. math:: \mathcal L = \frac{1}{2} \partial^\mu \phi \partial_\mu \phi - \frac{1}{2} m^2 \phi^2, one can easily prove that the corresponding energy-momentum tensor is .. math:: T^{\mu\nu} = \partial^\mu \phi\partial^\nu \phi - g^{\mu\nu} \mathcal L. .. admonition:: Derivation of Energy-momentum for Real Scalar Lagrangian :class: toggle QED. The 00 component is in fact the Hamiltonian density :math:`\mathcal H`. .. admonition:: Prove that :math:`T^{00}=\mathcal H` :class: toggle Calculate :math:`T^{00}`, .. math:: T^{00} =& \partial^0 \phi \partial^0\phi - \mathcal L \\ =& \frac{1}{2} ( \partial^0\phi\partial^0\phi + \partial^i \phi \partial^i\phi + m^2\phi^2 ). Notice that the Hamiltonian density is .. math:: \mathcal H = \Pi \partial^0 \phi - \mathcal L, where .. math:: \Pi = \frac{\partial \mathcal L}{\partial (\partial^0 \phi)} = \partial^0\phi. Plug in the momentum we find .. math:: \mathcal H = \partial^0\phi\partial^0\phi - \mathcal L = T^{00}. .. admonition:: Dialation and Noether Current :class: note Dilation can be written as .. math:: x_\mu \to & a x^\mu,\\ \phi \to & a^{-1} \phi. The Noether current corresponding to such transformation is .. math:: j_{\mathrm D}^{\mu} = T^{\mu\rho}x_\rho + \frac{1}{2} \partial^\mu \phi^2. Notice that Lagrangian .. math:: \mathcal L = \frac{1}{2} \partial_\mu \phi \partial^\mu \phi - \frac{1}{4\,! } \lambda \phi^4, which is :math:`\phi^4` theory, is invariant under dilation. References and Notes ----------------------------------------------------