So far not much had been said about connection to experimental evidence. Does difference of the (A0T d) and the (A0T p) variants of reading (A’) of physical possibility allow to distinguish them empirically? For ill posed problems (A0T d) allows physical possibilities which (A0T p) disavows; could we not simply decide between the variants by experimental means?
Testing the difference may seem straightforward. Let’s call s a treacherous state if s belongs to a state spaceE which renders the initial value problem ill posed, the initial value problem forsinEhas at least one solution, andsdoes not belong to any other Banach space that renders the initial value problem well posed. Successfully engineering a physical system whose initial configuration is represented by a treacherous state s could be evidence for the (A0T d) variant as this variant allows s to represent a physically possible state while (A0T d) does not. Impossibility of engineering a physical system described by any of the treacherous states could similarly be taken as evidence for the (A0T p) variant.
The proposed test faces several problems. First we would need to make sense of the condition that s does not ‘belong’ to any other Banach space that renders the initial value problem well posed. This condition presumes that an identification of states of different Banach spaces on the count that they represent the same physical configurations is available.
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If this physical identification of states is achieved, as it is frequently assumed, by a norm-preserving isomorphism then in the physically most relevant cases (when the Banach spaces are separable infinite dimensional Hilbert spaces) such identification always exists for all elements. Hence there are no examples of states s that ‘belong’ to some of the state spaces but do not belong to another and hence there are no treacherous states. The test then can not even get off the ground13.
Even if a treacherous state s existed in its state space E we would still face serious difficulties in the engineering part of the test. Many worries we raised in Chapter2about the necessity and sufficiency of well posedness for prediction and confirmation could be reiterated;
we only bring up here the issue of measurement inaccuracy. Recall that one of the empirical limitations typically attributed to observers is that they are only able to prepare and measure physical systems inaccurately. This inaccuracy gets a mathematical representation by means of closeness in the norm. The norm of spaceE is either operationally significant or it is not operationally significant14. If the norm of E is not operationally significant then it does not allow to correlate a set of close states with the prepared systems. If the norm E is operationally significant we still have the problem that any actual measurement has finite precision and hence our handicapped observers are never in the position to verify whether the prepared configuration is indeed appropriately represented by the treacherous state or by some other non-treacherous state close to the treacherous state15. The same problem would surface if we wanted to empirically test whether initial value indeterminism genuinely reflects a property of our world: such test would presumably require observers to repeatedly prepare a treacherous state s that gives rise to multiple solutions in E and check whether
13Physical identification of states of different Banach spaces on the basis of norm preserving isomorphism may however be problematic as we pointed out in Chapter2.
14A norm is operationally significant if states are sufficiently close in the norm iff their measurable prop-erties are also sufficiently close. See Chapter2for a discussion of operational significance and approximate measurability of the norm.
15If non-treacherous states are dense inE this problem can not be avoided.
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indeed more than one of the solutions can get realized. Non-uniqueness implies failure of continuous dependence and without continuous dependence it would be difficult to discern whether multiple solutions arise from the same treacherous state s or from different states nearby s16.
Testing failure of continuous dependence seems more promising. One could prepare many systems whose initial configurations do not differ more than a pre-set measurement precision and track the behavior of these systems. For an operationally significant norm such a close set of configurations can be mapped to a close set of states; the theory then could be used to estimate the maximal deviation of the solutions which (in case the norm is also approximately measurable) can be compared with the behavior of the system. A system that exhibits a deviation falling beyond the upper bound for the deviation could be taken as evidence for the failure of the (A0T p) variant.
One problem with such test could be that estimations of maximal deviation are tied to particular choices for the space E; typically there are multiple choices of E that can make the initial value problems well posed and it may not be the case that a universal upper bound exists that would be valid for all such possible E’s17. If there is no universal upper bound then the existence of a large deviation would not evidence failure of (A0T p) as it may only be an indication that we have not chosen the right space E to begin with18.
A more serious problem with this test, and with any other similar proposals, is that in order to draw any conclusion from experiments first we would need to be able to identify the norm or norms that are operationally significant and/or approximately measurable. Without a link connecting observed deviations with distances of states relating any experimental data
16Albeit see our comments in Chapter2about failure of continuous dependence not implying ‘everything goes.’
17This problem may be moot but I’m not aware of mathematical results settling this question either way.
18The existence of several Banach spacesEthat can make the problem well posed also points to a difficulty shared by both variants: even though (A0T p) helps in reducing the number of possible candidates it also faces the problem of choosing among the rest.
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to the mathematical models is problematic. To my knowledge no systematic study have been made in correlating norms to available experimental techniques. Sans establishing such correspondence it is not possible to do justice to the issue of continuous dependence by means of observation and we need to apply great care when we intend to interpret mathematical examples of failure of continuous dependence empirically19. This remark also applies to other empirical means that have less of an operationalist flavor. Although it would be interesting to see a detailed overview of the possible ways to empirically assess the difference of the two variants, and without such a detailed overview it would be hasty to conclude that no empirical means are available to tell them apart, we leave this discussion with the sense that finding such means is beset by many yet-to-be resolved problems.
4.5 OBJECTIONS
Variant (A0T p) of physical possibility supplies an argument for well posedness in limiting the choice of the state space E to Banach spaces that allow for the existence of the propagator.
This is a formal requirement: it does not imply approximate measurability or operational significance of the norm of E. Albeit severing ties with empirical justification may seem as an advantage it also makes the approach suspicious. It may be possible, for instance, that the equation is well posed in a norm that entails the existence of the modality generating propagator equation but it is not well posed in the operationally significant norm. This would then pose difficulties for prediction based on and confirmation of the underlying physical theory, which is the complaint originally raised against ill posed equations.
In Chapter2we argued that well posedness is neither necessary nor sufficient for
predic-19To avoid circularity we should also not base judgement of operational significance on whether a given norm renders the differential equations well posed. It is difficult to overcome such temptation given the prejudice for well posedness in the practice of mathematical physics.
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tion and confirmation. Among other objections we pointed out that even though a theory may not be able to supply prediction with certainty it may still be able to supply proba-bilities with which certain outcomes get realized. Quantum mechanics is a straightforward example: it is well confirmed in its probabilistic assertions but it is unable to supply defi-nite predictions. Quantum mechanics then may also be an example that supposedly should have raised our suspicion according to the previous paragraph. The Schr¨odinger equation is well posed in the operationally not significant L2 norm20; if it were possible to define an alternative notion of ‘state’ and an operationally significant norm in terms of the measur-able quantities alone then, since no measurmeasur-able quantities fix the time evolution of these quantities uniquely, the initial value problems for this alternative notion of ‘state’ would not be well posed. Then quantum mechanics would be a theory for which there is an argument for well posedness even though the argument does not establish that it is the operationally significant norm in which the theory should be well posed.
The argument presented in this chapter for well posedness has many limitations. Albeit many fundamental differential equations in physics can be cast in the 4.1 form – i.e. the Maxwell equation, the Schr¨odinger equation, and the Dirac equation for free particles (see Appendix A for some details) – there are others, such as Einstein field equations, that can not; it remains to be seen whether similar ideas can be pursued for nonlinear cases where no such straightforward relationship between the existence of propagators and well posedness of the differential equation is available.
Another contentious issue is that equations of the form 4.1handle the time parameter in a distinguished way which is then inherited by the propagator equation. Partial differential equations, at least prima facie, do not need to distinguish the time parameter from other physical variables. Thus the preferential treatment of time cries out for justification and so
20See Chapter2for an argument why theL2norm is not operationally significant; for the well posedness result see AppendixA.
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does the implied assumption of instantaneous states. In special relativity we don’t have a preferred coordinate system; general relativity even have models not admitting a global time function21.
The preferential treatment of the time variable may be less problematic than it appears;
an increasing literature on the nature of time points out that many differential equations of physics do treat the time parameter distinctly and indeed some philosophers, such as Callender(2008) orSkow(2007), make use of this feature of differential equations to explicate what makes time special. Coordinate system dependence and abandonment of instantaneous states may also be less of an issue, although in order to preserve well posedness additional requirements, such as finite signal propagation speed, might also need to be motivated22.
The main weakness of the approach lies in the arbitrariness of the applied mathematical concepts. Mathematical physics treats propagators as natural mathematical objects; from a pragmatic or an aesthetic point of view propagators can be viewed as natural candidates for representing a fixed component of a theory. But if we make propagators responsible for the modal character of physical theories and if the issue of well posedness turns on the propagators then it is natural to ask for a motivation for their mathematical properties. A propagator is a strongly continuous group23. Both strong continuity (which asserts that a solution does not have ‘jumps’ and is not to be confused with continuous dependence on the data) and the group property are, in the end, assumptions about how trajectories generated by the propagator behave. Any justification that can be given for such assumptions could likely directly act as a justification for well posedness of the differential equations.
I don’t know how this objection of petitio principiicould be conclusively preempted. We
21Note that models of general relativity that do not admit a global time function do not admit a well posed initial value formulation either. The mathematical framework we use here for abstract differential equations is not directly capable to handle differential equations of general relativity.
22The assumption of finite propagation speed is typically added to the notion of well posedness in general relativistic contexts, see i.e. (Wald;1984, p. 224)).
23See definitions in AppendixA
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have seen above that treating propagators as the modality generating fixed component brings some philosophical advantages – allowing for a truce between Governing and Non-Governing intuitions about laws, allowing interpreting the space E as the space of physically possible
‘states,’ etc. – apart from entailing well posedness. These may or may not be sufficiently compelling to accept the propagators-as-laws point of view. Although it is not a recourse for a philosopher, history of physics is also a history of well chosen definitions that cleverly hide conceptual problems, and choosing propagators as representations of laws might be one such choice to be made.
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