求证量子电动力学（QED）不具有主动微分同胚不变性
Alright, let's break down this rigorous proof clearly—starting with critical definitions to set the stage, then moving through QED's structure and the impact of active diffeomorphisms.
1. Key Definitions Recap
- Passive Diffeomorphism Invariance: This is a formulation-level property—it's just relabeling coordinates on spacetime. Every sensible field theory can be written to satisfy this, since you're only changing how you name points, not the underlying physics.
- Active Diffeomorphism Invariance: This is a theory-level property. It means if you smoothly and reversibly shift every point on the spacetime manifold (mapping point $p$ to $\phi(p)$) and transform all dynamical fields to follow this shift, the theory's action/Lagrangian remains unchanged. General Relativity (GR) has this because the metric itself is a dynamical field that shifts with the diffeomorphism.
2. QED's Lagrangian Structure
QED is defined on a fixed flat Minkowski background spacetime—the metric $\eta_{\mu\nu}$ is an external, non-dynamical structure that never changes. Its Lagrangian density is:
$$
\mathcal{L}{\text{QED}} = -\frac{1}{4}F{\mu\nu}F^{\mu\nu} + \bar{\psi}(i\gamma^\mu D_\mu - m)\psi
$$
Where:
- $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ is the electromagnetic field strength tensor, with $A_\mu$ as the four-potential;
- $\psi$ is the Dirac spinor field, and $D_\mu = \partial_\mu + ieA_\mu$ is the covariant derivative (defined for the fixed background metric);
- $\eta_{\mu\nu}$ is the fixed Minkowski metric, which is not a dynamical field in QED.
3. Field Transformations Under Active Diffeomorphisms
Take an active diffeomorphism $\phi: M \to M$, mapping spacetime point $x \mapsto x' = \phi(x)$. For dynamical fields, active transformations require:
- The spinor field transforms as $\psi'(x) = \psi(\phi^{-1}(x))$ (intuitively, the field's value at the new point $x$ is the value it had at the original point $\phi^{-1}(x)$ before the shift);
- The vector potential transforms via the pushforward map: $A'\mu(x) = \frac{\partial x^\nu}{\partial x'^\mu} A\nu(\phi^{-1}(x))$.
Crucially: the fixed background metric $\eta_{\mu\nu}$ does not transform under this diffeomorphism. It's a static external structure, not a field we're shifting along with the spacetime points.
4. How the Lagrangian Changes Under Active Diffeomorphisms
Now let's compute the transformed Lagrangian density $\mathcal{L}'_{\text{QED}}(x)$:
- For the field strength term: $F'{\mu\nu}(x) = \partial\mu A'\nu(x) - \partial\nu A'\mu(x)$. Substituting the transformed $A'\mu$, we get extra terms from the Jacobian of $\phi$, since the fixed metric doesn't adjust to compensate for the spacetime shift.
- For the Dirac spinor term: The covariant derivative $D'\mu \psi'(x) = \partial\mu \psi'(x) + ieA'\mu(x)\psi'(x)$. The derivative $\partial\mu \psi'(x)$ becomes $\frac{\partial (\phi{-1})\alpha}{\partial x^\mu} \partial_\alpha \psi(\phi^{-1}(x))$, which doesn't match the structure of the original covariant derivative—QED's derivative is built for the fixed background, not arbitrary spacetime shifts.
The action of QED is $S = \int_M \mathcal{L}{\text{QED}}(x) \sqrt{|\eta|} d^4x$. After the active diffeomorphism, the action $S' = \int_M \mathcal{L}'{\text{QED}}(x) \sqrt{|\eta|} d^4x$ (since $\eta$ is fixed, the volume element doesn't change). But $\mathcal{L}'{\text{QED}}(x) \neq \mathcal{L}{\text{QED}}(x)$ for arbitrary smooth $\phi$ (only trivial shifts like translations/Lorentz transformations leave it invariant, which are a tiny subset of all active diffeomorphisms).
5. Core Conclusion
QED relies on a fixed, non-dynamical background metric. Active diffeomorphism invariance requires all relevant structures (including the spacetime metric, which is dynamical in GR) to shift with the spacetime points. Since QED has no dynamical metric to adjust to arbitrary smooth spacetime shifts, its Lagrangian and action are not invariant under general active diffeomorphisms. Thus, QED does not possess active diffeomorphism invariance.
内容的提问来源于stack exchange，提问作者Tom Clarke
