5.3 Physical derivative dispositions

5.3.1 Hamiltonians, wave functions and measurements

In quantum physics, energy (the total of the kinetic and potential energies) is represented by the Hamiltonian operator . This operator enters into the Schrödinger wave equation which governs the time-dependence of all quantum wave forms. It thus generates all time evolution and all fields of probabilities for measurement outcomes, as discussed in Section 4.7. The principal dynamics in quantum physics are specified by knowing what the initial state is and what the Hamiltonian operator is. This applies to quantum mechanics as it is practised, by using Born’s statistical interpretation and then naively saying that the quantum state changes after a measurement to one of the eigenstates of the measurement operator. (This is the much discussed ‘reduction of the wave packet’, which we can agree at least appears to occur.)

We may consider quantum physics in the following ‘realistic’ way. We have the Hamiltonian which has to do with total energy. It is somehow ‘active’ since it is an operator which operates on the wave function and changes it. The Schrödinger equation is the rule for how the Hamiltonian operator produces the wave function. This wave function is a probabilistic disposition (a propensity) for action, since its squared modulus gives a probability for different macroscopic outcomes of experiments, and since the wave function changes according to the specific outcome.

Such is the structure of quantum physics as it is practised, and we may observe a sequence of derivative dispositions in operation:

• Hamiltonian operator: the fixed disposition to generate the wave function by evolving it in time,
• wave function: the probabilistic disposition (a ‘propensity wave’) for selecting measurement outcomes, and
• measurement outcome: the final result.

It appears again that we have multiple generative levels with the set of Hamiltonian, wave function and selection event. Note also that the final result is the weakest kind of minimal disposition, which influences merely by selection, because it is a selection. It appears as the last of a sequence of derivative dispositions, as a kind of ‘bottom line’ if we want to include it within the framework of multiple generative levels.

Admittedly, reductionist tendencies may be applied. It may be denied that there are distinct measurement outcomes in any ontological sense and that they may only be approximately defined within a coarse-grained ‘decoherent history’. Advocates of the Many Worlds Interpretation or Decoherence theories take this view. Others such as Bohr take the opposite view: he holds that only the measurement outcome is real and that the Hamiltonian and wave function are calculational devices and nothing real. These conflicting views will be discussed in Section 6.4.

5.3.2 Virtual and actual processes

Taking a broader view of contemporary physics and its frontiers, we may further say that the ‘Hamiltonians, wave functions and measurements’ from above describe just the dispositions for a class of ‘actual processes’. The Hamiltonian is the operator for the total energy. It contains both kinetic and potential energy terms. However, we know from Quantum Field Theory (QFT) that, for example, the Coulomb potential is composed ‘in some way’ by the exchange of virtual photons. Similarly, we also know from QFT that the mass in the kinetic energy part is not of a ‘bare mass’, but is of a ‘dressed mass’ arising (in some way) also from many virtual processes. This reiterates my theme: the Hamiltonian is not a simple disposition, but in fact is itself derivative from some prior generative level. This generative level could be called that of ‘virtual processes’ in contrast to that of ‘actual processes’.

The class of virtual processes, as described by QFT, has many properties that are opposite to those of actual processes of measurement outcomes. Virtual events are at points (not selections between macroscopic alternatives), are interactions (not selections), are continuous (not discrete), are deterministic (not probabilistic), and have intrinsic group structures (e.g. gauge invariance, renormalisation) as distinct from the branching tree structure of actual outcomes. These contrasts suggest that virtual processes should be distinguished from actual events. The guiding principles have different forms. Virtual processes are commonly described by a fixed Lagrangian subject to a variational principle in a Fock space of variable particle numbers, whereas actual processes deal with the energies of specific observable objects leading to definite measurement outcomes. We may draw these levels, in combination with the previous three levels of ordinary quantum mechanics, as

The dotted lines here show some similarities between the corresponding parts of the two main generative levels and will be discussed more in Chapter 24.

It is clear in physics that virtual processes form simultaneous ‘levels’ in addition to the ‘level’ of Hamiltonians, propensities and measurements. This is because virtual processes are clearly occurring perpetually and simultaneously with Hamiltonian evolution, as they are necessary to continually ‘prepare and form’ the ‘dressed’ masses and potentials in the Hamiltonian. Dressed masses and potentials persist during Hamiltonian evolution. In atoms and molecules, virtual processes such as photon exchanges to generate the Coulomb potentials exist continuously as a kind of background for observable processes.

5.3.3 Pregeometry and the generation of spacetime

Field theories such as QFT still use a geometric background of spacetime, and there is currently much speculative work in quantum gravity research to determine how this spacetime arises. Wheeler (1974) started interest in ‘pregeometry’: the attempt to formulate theories of causal processes which do not presuppose a differentiable manifold for spacetime. His aim was to encourage speculation as to how spacetime might arise. The task has been taken as showing how spacetime may turn out to be a ‘statistical approximation’ in some limit of large numbers of hypothetical pregeometric processes. Proposals have involved spinors by Penrose and Rindler (1987); ‘loop quantum gravity’ as described, for example, in Rovelli (1998); and ‘causal sets’ according to Bombelli et al. (1987) and Brightwell et al. (2003).

If some pregeometry could be identified, I would speculate that a good way of understanding it would be as a distinct pregeometric level within a structure of derivative dispositions. That is, instead of spacetime being a statistical approximation (in the way thermodynamics is a statistical approximation to molecular gas theories), it could be better imagined that spacetime is an aspect of derivative dispositions that have been generated by ‘prior’ pregeometric dispositions. It is in keeping with Wheeler’s use of the word ‘arising’. This is speculative, but it does follow the pattern of some current research. I use it as an example of how the philosophical analysis of dispositions may interact fruitfully with modern physics. This appears to be useful particularly since the very aim of ‘deriving spacetime’ has itself been called into question by Meschini (2008).