# Space-Time Foliation and Frozen Formalism. Note Quote. The idea is that one foliates space-time into space and time and considers as fundamental canonical variables the three metric qab and as canonically conjugate momentum a quantity that is closely related to the extrinsic curvature Kab. The time-time and the space-time portions of the space-time metric (known as the lapse and shift vector) appear as Lagrange multipliers in the action, which means that the theory has constraints. In total there are four constraints, that structure themselves into a vector and a scalar. These constraints are the imprint in the canonical theory of the diffeomorphism invariance of the four-dimensional theory. They also contain the dynamics of the theory, since the Hamiltonian identically vanishes. This is not surprising, it is the way in which the canonical formalism tells us that the split into space and time that we perform is a fiduciary one. If one attempts to quantize this theory one starts by choosing a polarization for the wavefunctions (usually functions of the three metric) and one has to implement the constraints as operator equations. These will assure that the wavefunctions embody the symmetries of the theory. The diffeomorphism constraint has a geometrical interpretation, demanding that the wavefunctions be functions of the “three-geometry” and not of the three-metric, that is, that they be invariant under diffeomorphisms of the three manifold. The Hamiltonian constraint does not admit a simple geometric interpretation and should be implemented as an operatorial equation. Unfortunately, it is a complicated non-polynomial function of the basic variables and little progress had been made towards realizing it as a quantum operator ever since De Witt considered the problem in the 60’s. Let us recall that in this context regularization is a highly non-trivial process, since most common regulators used in quantum field theory violate diffeomorphism invariance. Even if we ignore these technical details, the resulting theory appears as very difficult to interpret. The theory has no explicit dynamics, one is in the “frozen formalism”. Wavefunctions are annihilated by the constraints and observable quantities commute with the constraints. Observables are better described, as Kuchar emphasizes, as “perennials”. The expectation is that in physical situations some of the variables of the theory will play the role of “time” and in terms of them one would be able to define a “true” dynamics in a relational way, and a non-vanishing Hamiltonian. In contrast to superstring theory, canonical quantum gravity seeks a non-perturbative quantum theory of only the gravitational field. It aims for consistency between quantum mechanics and gravity, not unification of all the different fields. The main idea is to apply standard quantization procedures to the general theory of relativity. To apply these procedures, it is necessary to cast general relativity into canonical (Hamiltonian) form, and then quantize in the usual way. This was partially successfully done by Dirac. Since it puts relativity into a more familiar form, it makes an otherwise daunting task seem hard but manageable.

# Matter Defined as Just Another Quantum State: Whatever Ontologies. In quantum physics, vacuum is defined as the ground state of a quantum field. It is a state of minimum energy, corresponding to zero particles. Note that this definition of vacuum uses already the conceptual and formal machinery of quantum field theory. It is justifiable to ask weather it is possible to give a more theory-independent definition with lesser theoretical load. In this situation vacuum would be an entity which is explained – not just defined within and then explored – by quantum field theory. For example, one could attempt an operational definition of vacuum as the state in which no particles are detected. But then we have to specify how to detect the particles, with what efficiency, etc., that is, we need a model for the particle detector. Such a model, known as the Unruh-DeWitt detector, is constructed however from within quantum field theory. Unruh-DeWitt detector is a simplified model of a real particle detector. Its basic property is the fact that it is linearly coupled to the field, so that it can detect one-particle states. Indeed, as long as the detector moves inertially in Minkowski spacetime, it really does react to one-particle states and not to the 0-particle state (vacuum). However, when it moves non-inertially, it may react even in the vacuum. The energy needed for the reaction in the vacuum comes from the agency that accelerates the detector (not from the vacuum energy). The vacuum is simply a special state of the quantum field – implying that quantum physics allows the return of the concept of ether, although in a rather weaker, modified form. This new ether – the quantum vacuum – does not contradict the special theory of relativity because the vacuum of the known fields are constructed to be Lorentz-invariant. In some sense, each particle in motion carries with it its own ether, thus Lorentz transformations act in the same way on the vacuum and on the particle itself. Otherwise, the vacuum state is not that different from any other wavefunction in the Hilbert space. Attaching probability amplitudes to the ground state is allowed to the same degree as attaching probability amplitudes to any other state with nonzero number of particles. In particular, one expects to be able to generate a real property – a value for an observable – in the same way as for any other state: by perturbation, evolution, and measurement. The picture that quantum field theory provides is that both particles and vacuum are now constructed from the same “substance”, namely the quantum states of the fields at each point (or, equivalently, that of the modes). What we used to call matter is just another quantum state, and so is the absence of matter – there is no underlying substance that makes up particles as opposed to the absence of this substance when particles are not present. One could even turn around the tables and say that everything is made of vacuum – indeed, the vacuum is just one special combination of states of the quantum field, and so are the particles. In this way, the difference between the two worldviews, the one where everything is a plenum and vacuum does not exist, and the other where the world is empty space (nonbeing) filled with entities that truly have the attribute of being, is completely dissolved. Quantum physics essentially tells us that there is a third option, in which these two pictures of the world are just two complementary aspects. In quantum physics the objects inhabit at the same time the world of the continuum and that of the discrete.

Incidentally, the discussion has implications for the concept of individuality, a pivotal one both in philosophy and in statistical physics. Two objects are distinguishable if there is at least one property which can be used to make the difference between them. In the classical world, finding this property is not difficult, because any two objects have a large amount of properties that can be analyzed to find a different one. But, because in quantum field theory objects are only combinations of modes, with no additional properties, it means that one can have objects which cannot be distinguished one from each other even in principle. For example, two electrons are perfectly identical. To use a well-known Aristotelian distinction, they have no accidental properties, they are truly made of the same essence.

To see in a simple way why quantum physics requires a re-evaluation of the concept of emptiness, the following qualitative argument is useful: the Heisenberg uncertainty principle shows that, if a state has a well-defined number of particles (zero) the phase of the corresponding field cannot be well-defined. Thus, quantum fluctuations of the phase appear as an immediate consequence of the very definition of emptiness. Another argument can be put forward: the classical concept of emptiness assumes the separability of space in distinct volumes. Indeed, to be able to say that nothing exists in a region of space, we implicitly assume that it is possible to delimitate that region of space from the rest of the world. We do this by surrounding it with walls of some sort. In particular, the thickness of the walls is irrelevant in the classical picture, and, as long as the particles do not have enough energy to penetrate the wall, all that matters is the volume cut out from space. Yet, quantum physics teaches us that, due to the phenomenon of tunneling, this is only possible to some extent – there is, in reality, a non-zero probability for a particle to go through the walls even if classically they are prohibited to do so because they do not have enough energy. This already suggests that, even if we start with zero particles in that region, there is no guarantee that the number of particles is conserved if e.g. we change the shape of the enclosure by moving the walls. This is precisely what happens in the case of the dynamical Casimir effect. These demonstrate that in quantum field theory the vacuum state is not just an inert background in which fields propagate, but a dynamic entity containing the seeds of multiple possibilities, which are actualized once the vacuum is disturbed in specific ways.