Actividades

If I pick up a rock, hold it in my hand and then release it, it will fall to the ground. If instead of performing that sequence of actions today, I do them tomorrow, then the rock will still fall to the ground in the same way. If, 500 years from now, one of our robot overlords does those actions, the rock will still fall to the ground. It does not matter where I stand before releasing the rock. The rock will move in the same way regardless of whether I stand here or at the end of the street. Even moving somewhere more exotic will not change that; a rock released on the surface of Mars will still fall towards the (Martian) ground. This may not be a motion towards the

surface of Earth as in the previous cases, but it still fits the general pattern. Objects with mass experience forces attracting them to one another in a predictable way formalised by Newtonian mechanics (of course there are certain limits to this, I am ignoring General Relativity for expositional convenience).

None of that will come as a surprise. It is a familiar thought that objects move in the same sort of way no matter where or when they happen to be. As Wigner noted, the spatiotemporal invariance of laws looks to be a requirement on our being able to discover them.122 _{After all, if there was no reason to think that experiments}

performed at different times would get the same results, then experiments would no longer be repeatable. There would be no way to disconfirm someone’s claims, since the conditions under which I perform an experiment might fit into a different regularity to the conditions under which you perform your experiment. Further, if there was enough spatiotemporal variation of the laws then there would be little point in attempting any form of ordered investigation into the nature of the world. If we did not believe that there were universal patterns to be discovered, there would be little point in developing any kind of scientific method.

The modern notion of a symmetry is a development of a concept that has been in use for a very long time. Informally, we might say that something possesses a symmetry if it remains unchanged by a certain action. A classic example would be of a square. Rotating a square through an angle that is a multiple of 90° leaves the square the same as it was pre-rotation. Similarly, reflecting the square either through one of its centre lines or one of its diagonals will not affect it. These eight transformations that do not affect the square are referred to as its symmetries. That they form a mathematical group leads to the modern definition: something possesses a symmetry when it is invariant under a certain group of transformations. The generality of this definition means that it is not just physical objects that can have symmetries. Laws too can exhibit symmetries when they remain unchanged under some transformation. The laws of Newtonian mechanics are invariant under

122 _{Wigner (1949) makes this point. Brading and Castellani’s introduction to their (2003) also}

mentions that this is sometimes seen as a prerequisite to describe the world by modern science.

spatial and temporal translations as we have just rehearsed. These two are not the only spacetime symmetries. For example, it does not matter how my rock is oriented in space when I release it, which is just to say that space exhibits rotational invariance. Nor are the laws associated with spacetime the only ones which possess symmetries. Consider Coulomb’s law:

𝐹 = 𝑘𝑞1𝑞2 𝑟2

Coulomb’s law claims that the force between two charged particles is proportional to the charges on the particles and inversely proportional to the square of the distance between them. None of this is dependent on when the two particles exist, the lack of this dependence implies that Coulomb’s law is temporally invariant. Similarly, it is the distance between the particles that matters for the magnitude of the force exerted, not their absolute positions. As a spatial translation of such a system will not change anything about it, Coulomb’s law is also spatially invariant. Giving the mathematical form of Coulomb’s law makes it clear that the law is invariant under inversion of charges. If q1 and q2 both had their signs inverted, the

force between the two particles would remain the same. This corresponds to the truth of an accompanying counterfactual: had all the charges in the world been inverted, there would have been no empirically discernible difference.

Some of these symmetries are tied closely into how we normally think about laws. One common expectation regarding fundamental laws is that they be universal such that they do not hold only in one restricted area of spacetime. But while we expect laws to accord with this sort of symmetry, they are not conceptually required to do so.123 _{The requirement mentioned above, that their doing so is necessary for modern}

science, is an epistemic requirement. It is not an ontic one. That we need the laws to be spatiotemporally invariant to make an ordered investigation into the world does not mean that there would be no laws if this sort of invariance failed. Rather, it would simply be more difficult – impossible perhaps – for us to find out what they would be. We might be able to cope with a lack of invariance if it made a difference only in

123 _{This expectation that the laws take a certain form is a demonstration of the heuristic role}

extreme situations or if it manifested in a particularly ordered way, but this is an epistemic requirement brought about by our epistemic limitations.

Lange gives a more concrete example of a world where not every law is invariant under spatial translations.124 _{Suppose that some world has a privileged centre and}

that every object at this world experiences a force towards this centre inversely proportional to the square of the separation of object and centre-point. The world also has a law that each object experiences this force. The law at such a world will not demonstrate spatial invariance. A translation of everything in that world’s space that leaves the forces and centre-point unchanged will result in a world where the associated ‘law’ is false. Since we can imagine a world in which spatial displacement makes a difference to the laws, invariance under spatial translation cannot be a conceptual requirement on laws.

Investigation into the relationship between the world and symmetries is often not done from the armchair as in the case of Lange’s world with the privileged centre. Experimental results are expected to accord with those symmetries that we think hold at our world, but this does not always happen. A recent case of this was in 2011, when a team led by Webb published their (second) findings on fine structure constant variations.125 _{The fine structure constant, typically denoted by α, is a}

measure of the strength of electromagnetism. By analysing the light coming in from distant quasars, the team found indications that this ‘constant’ has a lower value in one direction and a higher one in the other, suggesting a spatial dipole. If the immediate interpretation of the results is correct, then we have evidence that the laws are not the same everywhere. Even if an error is found in either the experiments or the interpretation of the data, this is a clear indication that working scientists are willing to take seriously the suggestion that one of the classic spacetime symmetries fails. A note of caution here on just what is varying. In discussions of possible worlds with different laws to our own, it is common to see examples like an inverse cube law of gravity as opposed to an inverse square. In those cases, it is the form of the

124 _{Lange (2007) p. 461.} 125 _{Webb et al. (2011).}

law that is varying (albeit to a minor degree).126 _{This sort of variation is not supported}

by the recent experiments. Rather, the variation they suggest is in the constants that occur in the laws. This is still variation, of course, but of a much more restricted kind to that usually seen in philosophical imagination.

A more famous case is concerned with mirror reflection. Up until the mid-1950s, physicists thought that nature had no way of distinguishing between ‘right- handedness’ or ‘left-handedness’ (more commonly referred to as ‘parity’).127 _This

was the belief that the laws of nature were symmetric under mirror reflection. We can define right or left relative to ourselves and our surroundings but there is no intrinsic difference between right and left hands.128 _{Hence a mirror reflection of the}

world would make no difference. A series of experiments showed that this is not the case for weak decay and that there is a difference between mirror reflections. At small enough scales, parity breaks down. It turns out that even commonsensical symmetries are vulnerable to empirical disconfirmation.

The situation is somewhat more complicated than just indicated, as these things often are. Parity is one of the symmetries that goes into the so-called CPT Theorem (corresponding, naturally, to the P). The letter C stands for charge-conjugation, the operation of swapping particles out for their corresponding antiparticles (in the quantum context, this is more impactful than the mere swapping of charge signs indicated earlier). Here, T is the time-reversal operation, the temporal analogue of P’s spatial inversion. The product of these three symmetries, the CPT-symmetry, is taken to imply that a mirror image of our world would evolve in the same way. While the C, P and T symmetries might be individually violated, the CPT-symmetry is currently taken by the Standard Model of particle physics to be one that holds at our world.129

126 _{The conservative nature of these changes is noted by McKenzie (2014).} 127 _{Dardo (2004) pp. 260-264.}

128 _{For a defence of the lack of intrinsic differences, see Pooley (2003). For commentary, see}

Huggett (2003).

The symmetries of interest to physics are commonly classified in some of the following ways.130 _{First there are the spacetime symmetries, examples of which we}

have already encountered. These are contrasted with the symmetries which do not involve spacetime (such as permutation invariance, discussed below). There are global symmetries such as rotation, those whose transformations can be specified in a time-independent way. These should be contrasted with local symmetries, those whose transformations are time-dependent.131 _{Some symmetries are continuous –}

their transformations can come in arbitrary amounts (think of the possible rotations of a circle; any degree of rotation leaves the circle unaffected). Other symmetries are discrete – the associated transformations are not continuous in this way (think of the possible rotations of a square; only rotations that are a multiple of 90° leave the square unaffected). More recently, Caulton has argued that it is useful to distinguish between ‘analytic’ symmetries and ‘synthetic’ ones. Only the holding of the latter corresponds to physical differences; the former are to be explained as constraints on theory interpretations.132

Symmetries are often claimed to play particularly important roles. One of these is philosophical: that certain ones obtain is supposed to tell us something about the nature of the world. The classic example here is permutation invariance, which is often taken to mean that quantum particles are not individuals.133 _{This is typically}

introduced by way of a rough analogy. Suppose that we have two balls (corresponding to particles) and two boxes (the microstates that the particles can be in). Classically, there are four different ways to distribute the balls across the boxes: both in the left box, both in the right, one in each and finally the other way of putting one in each. Assuming that each of these distributions is equally likely, the chance of any one of them is 1/4. This leads to an empirical consequence: if there were some

130 _{Accessible overviews are provided by Bangu (2013) and Morrison (2008).}

131 _{For a discussion of why it is that time-independence has priority in this definition, see}

Wallace (2003). Earman (2003) complains that much discussion of this distinction is misleadingly presented.

132 _{Caulton (2015).}

133 _{Here I draw upon French and Rickles (2003), and French and Krause (2006). The following}

process that randomly assigned balls to boxes, we would expect the frequency of each distribution to tend towards 1/4 as time passed.

The classical case is straightforward enough because permuting the balls gives rise to new states of affairs. The quantum case is more interesting because permutation invariance is taken to imply that permuting balls (or, more accurately, particles) does not lead to a new possibility. Interchanging them produces no observable difference. Combinations of quantum mechanical particles have two forms of statistical behaviour. Bosons act according to Bose-Einstein statistics while fermions act according to Fermi-Dirac statistics. Relating this to the example set-up, since bosons can be in the same quantum state, bosonic balls can occupy the same box. On the other hand, Pauli’s Exclusion Principle rules out fermions from being in the same state. Fermionic balls, therefore, cannot reside in the same box. The difference in statistical behaviour is associated with a difference in the relevant possibilities. When distributing balls that obey Bose-Einstein statistics, there are only three possible distributions: both balls to the left, both to the right and one in each box. We ‘lose’ a possibility by moving away from the classical case since, from the theory’s perspective, the distributions where each ball has its own box are identical to one another (this is the invariance: permuting the balls makes no difference). The situation is even more stark for the fermionic balls. Once we have made the same identification as in the case of the bosons and ruled out distributions where both balls are in the same box, we have only a single distribution left: a ball in the left box and a ball in the right. Consequently, the probabilities in the two quantum cases differ from that in the classical case. The probability of finding any one configuration of bosons (assuming all the distributions are equally likely) is 1/3. Given that there is only one possible configuration of fermions, it will receive a probability of 1.

The philosophical import of this is found in the identification of permuted configurations. One might take the fact that permutations of classical particles are treated as new arrangements to indicate that the particles have some form of identity that goes beyond their intrinsic properties (basing their individuality on their having different intrinsic properties fails as all instances of each kind of particle has these properties in common). This might be some form of haecceity or primitive

substance. But it is not required that we invoke metaphysically loaded notions like these. If we assume that some principle of impenetrability holds such that no two objects occupy the same spatiotemporal point, then classical particles will differ from one another in their spatiotemporal relations. Empiricists sceptical of primitive individuality might therefore wish to ground the individuality of particles in these different relations. This is an appeal to a Principle of the Identity of Indiscernibles, a claim that any two distinct objects must be distinguishable (that is, they must differ in their properties or relations).

However, since permutation does not lead to any empirical difference when we are concerned with quantum statistics, the status of such a principle is more controversial when applied to quantum particles. Two electrons, for example, entangled in the singlet state will possess the same intrinsic properties as each other (these discussions usually make implicit appeal to the so-called Eigenstate- Eigenvalue link, discussion of which I set to the side as tangential). Since quantum particles do not generally have unique spatiotemporal trajectories, at least under the standard interpretation of quantum theory, we cannot appeal to different spatiotemporal relations to distinguish them as we can for classical particles. This is commonly taken to indicate that the particles under discussion lack individuality in some substantive way. The evocative metaphor here is that particles are akin to money in the bank. If I have £100 in my account and withdraw £10, it makes little sense to ask which 10 of the original 100 that I took. The pounds in a bank account lack the individuality required to ask those sorts of questions about them.

There are moves that can be made in response to this.134 _{Saunders has suggested a}

particularly prominent response, reviving Quine’s three grades of discriminability and arguing that even the individuality of entangled particles can be grounded in the fact that there exists a non-reflexive relation between them: an electron in the singlet state bears the relation of having opposite spin to another electron, but not to itself. This strategy has the advantage of covering other philosophically well- known cases, such as Black’s two spheres.135 _{It is not maximally applicable, however,}

134 _{See French and Krause (2006) pp. 149-173 for discussion of these.} 135 _{Black (1952).}

since it will not suffice to recover a Principle of the Identity of Indiscernibles in every domain. Graph theory, despite sometimes being thought to provide cases appropriately analogous to quantum particles, has edgeless graphs which contain vertices that do not satisfy such a principle.136 _{On a more physical note, whether}

bosons are weakly discernible is still a contentious matter of debate.

Of course, it is not just philosophers who are interested in the holding of symmetries; they are enormously important to the practice of physics. The acceptance of Einstein’s work on relativity marked, to use Wigner’s phrase, ‘the reversal of a trend’.137 _{Whereas laws had taken centre place in our efforts to understand the}

world, now principles of invariance were of prime importance. Years later, this led Nobel laureate Philip Anderson to comment that ‘It is only slightly overstating the case to say that physics is the study of symmetry.’138

An indication of this importance is found in the use of symmetry groups to predict phenomena. Briefly, this is where a certain kind of invariance principle is posited and then particle behaviour is predicted on the assumption that the world behaves according to such a principle. The famous historical example of this is the Eightfold Way and the completion of the spin-3/2 baryon decouplet. When classifying particles, it is common to fit them into particular families called multiplets. The