# Must a classical Lagrangian or a Hamiltonian be a real function?

$$\bullet$$ Is it fair to assume that the classical Hamiltonian or Lagrangian of a system (a particle or a field) is always a real-valued function?

$$\bullet$$ If not, can you provide counter-examples?

$$\bullet$$ The cases in which the Hamiltonian represents the total energy of a system, it must be real on physical grounds. Apart from that is there any other criterion which demands that a Lagrangian or Hamiltonian must be real?

• Using your comment about the Hamiltonian and Energy as a starting point, the Lagrangian is essentially the Laplace transform of H (or -H). So, under normal circumstances L would be real too. Unless you added some imaginary degrees of freedom to the configuration space. Another question might be whether you could add non-real dof to the system while maintaining a real valued H and L.
– user196418
Nov 4, 2018 at 22:04
• possible duplicate: Hermiticity of the Lagrangian in QFT. Nov 4, 2018 at 22:24
• Possible duplicates: physics.stackexchange.com/q/127797/2451 , physics.stackexchange.com/q/46528/2451 and links therein. Nov 4, 2018 at 22:24

Any physical lagrangian or hamiltonian is real. There may be models out there in which $$\cal{L}$$ and $$\cal{H}$$ are complex, but in those cases the "complexity" must be for mathematical convenience, and the quantities calculated in such a model will need to be converted to real quantities or simply not correspond to physical observables. One strong reason for this is that the integral of a complex function is, in general , complex as well. Therefore a complex lagrangian would give rise to a complex action, which doesn't make any sense. Even if the hamiltonian doesn't correspond to the energy, a complex-value $$\cal{H}$$ would result in a complex lagrangian, so the problem remains.
Example: let us note that the standard Dirac Lagrangian $$\bar{\psi}(i\gamma^\mu\partial_\mu-m)\psi$$ is generally complex (and you can consider this theory classical before second quantization), but you can add a complex total divergence and make this Lagrangian real (symmetric Lagrangian $$\frac{i}{2} (\bar\psi \gamma^\mu \partial_\mu \psi - (\partial_\mu \bar\psi) \gamma^\mu \psi ) - m \bar\psi \psi$$).