When are the different choices of spacelike hypersurfaces equivalent for the quantisation of a field theory?

Question

Suppose we have a field theory with a Lagrangian density $L$ on a Lorentzian manifold $(M,g)$. For the specific case of Minkowski spacetime, we can impose the equal-$t$ quantisation CCR:

$$[\phi (x), \frac{\partial L}{\partial\partial _t\phi }(y)]=i\hbar \delta(x-y)];\qquad [\phi(x), \phi(y)]=0.$$

To do this, we first choose a spacelike hypersurface corresponding to any of the inertial frames. Then the points $x,y$ refer to points on that hypersurface. We want the quantum theory to be independent of the choice of hypersurface chosen for quantisation.

We know that, for this specific case where we consider Minkowski spacetime and inertial frames, the choice of hypersurface doesn't matter. As in, we can impose it in any of the inertial frame space-like hypersurfaces and it automatically holds for the other inertial frame spacelike hypersurfaces as a consequence of the Heisenberg picture dynamics. So the hypothesis we have is that :

$$\text {CCR on arbitrarily chosen hypersurface } + \text{Euler Lagrange equations}\implies \text{CCR on any other hypersurface}$$

How general is this result? I will try to phrase the problem generally. Generally, we are looking at a field theory on a time orientable Lorentzian spacetime $(M,g)$ having a Lagrangian density $L$. We pick an arbitrary space-like Cauchy hypersurface. The co ordinates on it are $x^i, i=1,2,3$. Let $v^{\mu}(x_i)$ be a future directed time-like vector orthogonal to the hypersurface at point $x^i$, such that $g_{\mu \nu} v^{\mu} v^{\nu}=-1$ (using $-+++$ metric). Then we impose:

$$[\phi(x^i), v^{\mu}(y^i) \frac{\partial L}{\partial(\partial ^{\mu} \phi)} (y^i)]=i\hbar \delta ^3(x^i-y^i)$$ $$[\phi (x^i), \phi (y^i)]=0$$

Now, we want the above to be independent of the choice of hypersurface. So we want to check if the above condition automatically holds for all the other hypersurfaces if we impose it on any of the hypersurfaces.

What are the conditions on $M,g,L$ for this to hold?

I think the second CCR $$[\phi (x^i), \phi (y^i)]=0$$ is the causality condition for space-like separated observables. So I think this imposes a condition on $L$ that the dynamics must not allow FTL. In Minkowski spacetime, this condition requires $L$ to be Lorentz invariant. I'm not sure what this condition is in general.

What are the general conditions on $M,g,L$?

Answer 1

The standard conditions are:

(M,g) must be globally hyperbolic, i.e., g is Lorentzian and there exists some foliation into Cauchy surfaces. And the action defined by L must be (at least) Poincare invariant.

Theories on different Cauchy surfaces are related (on the formal level) by Tomonaga-Schwinger equations, which are essentially a quantum formulation of the classical covariant formulation on the space of solutions. For the latter, see, e.g.,

J. Lee and R.M. Wald, Local symmetries and constraints, J. Math. Phys. 31 (1990), 725-743.

The main point is that the symplectic form is the same independent of the Cauchy surface, hence the quantizations should be equivalent. But this is not a rigorous result since very litlle of interacting QFT in curved spacetimes is rigorous.

Answer 2

The standard reference for the math of curved spacetime QFT is Wald's 1994 book "Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics", especially chapter 4 where he goes over these issues in some detail. The full details are too lengthy for one answer, but if you can obtain chapters 3 and 4 of that book, you can see for yourself. In brief, let

$$ (\nabla^\mu\nabla_\mu - m^2)\phi (x)=0 $$

be the curved spacetime Klein-Gordon operator, i.e. $\nabla ^\mu$ is the derivative compatible with $g$. Then

• If $(\mathcal{M},g)$ is globally hyperbolic with Cauchy surface $\Sigma$, it can be foliated by a one-parameter family of smooth Cauchy surfaces $\Sigma _t$ and has topology $\mathbb{R}\times \Sigma$ (Geroch 1970; Dieckmann 1988).
• For spaces satisfying the above, the Klein-Gordon problem is well-posed and has an unique solution on all of $\Sigma$.

A point in the classical phase space is then given by the pair $(\phi, \pi)$, each of which gives rise to a unique solution as per the above theorem (4.1.2 in Wald's book). This also implies that which hypersurface is chosen as the "initial" time is irrelevant.

The classical theory is then well-behaved for a fairly broad range of classical theories and Klein-Gordon equations with a source term. The quantum theory, less so. To move to a quantum field theory, one has to specify a symplectic form $\Omega$:

$$ \Omega ([\phi _1, \pi _1],[\phi _2, \pi _2]) = \int_{\Sigma _0} d^3x [\phi _2 n^a\nabla _a\phi _1-\phi _1n^a\nabla _a \phi _2]\sqrt{h} $$ with $n^a$ the time-like vector and $\Sigma _0$ the space-like Cauchy surface.

One then proceeds to construct a bilinear form $\mathcal{S}\times \mathcal{S}\mapsto \mathbb{R}$ from the space of solutions $\mathcal{S}$ to the Klein-Gordon equation such that

$$ \mu (\psi _1, \psi_1) = \text{l.u.b}\frac{1}{4}\frac{[\Omega (\psi _1,\psi _2)]^2}{\mu (\psi _2,\psi _2)} $$ with l.u.b = least upper bound.

This is taken as the inner product of the theory, and with a little calculation one can show that it allows you to define a complex structure on the space of solutions which allows the definition of the positive and negative energy solutions (thus creation and annihilation operators). This choice is not unique in general. The conditions are as follows:

Take two such bilinear forms $\mu _1$ and $\mu _2$. Then the two theories defined by these are unitarily equivalent if

1. The operator Q in the space of solutions defined by $\mu _1 (\psi _1, Q\psi _2) = \mu _2 (\psi _1, \psi_2) - \mu_1 (\psi _1,\psi _2)$ is of trace class.
2. There exists $C,C'>0$ such that $C\mu _1 (\psi , \psi ) \leq \mu _2 (\psi , \psi ) \leq C'\mu _1 (\psi , \psi)$ for all $\psi \in \mathcal{S}$.

(theorem 4.4.1 of Wald's book). If these and the previously mentioned conditions on the spacetime hold, the two quantum field theories defined by $\mu _1$ and $\mu _2$ are unitary equivalent. So if you take two Cauchy surfaces and construct the creation operators, you can check unitary equivalence of your two theories by these conditions. That's the most general result I'm aware of.