In classical field theory, we consider the Lagrangians with single time-derivative of fields whereas double derivative of the field w.r.t. space is allowed sometimes. I understand that the reason of abandoning the 2nd order time-derivative of the fields is that we require two initial conditions, one is that of the field and the second is that if the momentum of the field.
What I don't understand is what is the problem with specifying the two initial conditions?
Also, while moving over to QFT from the classical description, how come the above mentioned discrimination of time derivative over space derivative, does not contradict the notion of putting space and time on equal footing?
Answer
Metric signature convention: $(+---)$.
First, note that physical dynamics is ultimately decided by the equations of motion, which you get from the Lagrangian $\mathcal{L}$ after using the least action principle. The kinetic term in a $1$-derivative (before integration by parts) field theory goes like $\mathcal{L} \sim \partial_\mu \phi \partial^\mu \phi \sim -\phi \square \phi$ whose equations of motion are $\square \phi + \cdots = 0$. This is a second order differential equation and so needs two initial conditions if you want to simulate the system.
The reason why people get nervous when they see higher derivatives in Lagrangians is that they typically lead to ghosts: wrong-sign kinetic terms, which typically leads to instabilities of the system. Before going to field theory, in classical mechanics, the Ostrogradsky instability says that non-degenerate Lagrangians with higher than first order time derivatives lead to a Hamiltonian $\mathcal{H}$ with one of the conjugate momenta occurring linearly in $\mathcal{H}$. This makes $\mathcal{H}$ unbounded from below. In field theory, kinetic terms like $\mathcal{L} \sim \square \phi (\square+m^2) \phi$ are bad because they lead to negative energies/vacuum instability/loss of unitarity. It has a propagator that goes like $$ \sim \frac{1}{k^2} - \frac{1}{k^2-m^2}$$
where the massive degree of freedom has a wrong sign. Actually, in a free theory, you can have higher derivatives in $\mathcal{L}$ and be fine with it. You won't 'see' the effect of having unbounded energies until you let your ghost-like system interact with a healthy sector. Then, a ghost system with Hamiltonian unbounded from below will interact with a healthy system with Hamiltonian bounded from below. Energy and momentum conservation do not prevent them from exchanging energy with each other indefinitely, leading to instabilities. In a quantum field theory, things get bad from the get-go because (if your theory has a healthy sector, like our real world) the vacuum is itself unstable and nothing prevents it from decaying into a pair of ghosts and photons, for instance.
This problem of ghosts is in addition to the general consternation one has when they are required to provide many initial conditions to deal with the initial value problem.
Also, in certain effective field theories, you can get wrong-sign spatial gradients $ \mathcal L \sim \dot{\phi}^2 + (\nabla \phi)^2$. (Note that Lorentz invariance is broken here). These lead to gradient instabilities.
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