Effect of Extra Dimensions and Parity Violation in Externalist vs Internalist debate of Incongruent Counterparts
Discussion of whether the space is an absolute entity or a relative phenomenon due to other objects or matter is over a millennium old. One interesting argument weighing in on this problem is presented using the analysis of incongruent counterparts. As Kant argued, existence of identical mirror images of asymmetric objects such as left and right hands implies that the substantivalist viewpoint of space is correct. Relativists rejection of Kant’s argument has given rise to two different explanation of the phenomenon. First of these, internalism, argues that the handedness of objects is an intrinsic property. Externalists, on the other hand, claims that the differences only materialize when comparing two objects, and deny any intrinsic handedness[1, p12].
I will analyze two arguments related to the debate between internalism and externalism: namely, the affect of extra dimensions and fall of parity and whether they strengthen or weaken the above positions. I think the arguments related to extra dimensions do not bare any strength for or against either of the positions. I would also try to show that while externalism is certainly questioned by the fall of parity, it is fairly easy to defend externalist position as shown by[2]. I will also very briefly discuss substantivalism as a much stronger position than either of these relativist positions with the advent of quantum field theory.
It is important to clarify the difference between the two main positions in support of relativism. Internalists argue that there is inherent differences between incongruent counterparts, and that their left or right orientation is explainable without appealing to any external entity. For instance, it is possible to argue that three dimensional asymmetry in any object implies an orientation and an incongruent counterpart. Externalists, on the other hand, reject this idea and agree with Kant that a given hand’s orientation is not describable by distance relations between its own sub parts. They argue that the differences in incongruent counterparts lies in their distance relations to external objects, especially to their own counterparts. A common view is that the differences are explained mathematically by using two counterparts and the distances to each corresponding point. This account use only relational accounts, and can explain why the two objects are different1[1].
Internalist account of similarity of mirror asymmetric objects face few critical challenges. One of the most prominent challenges is the possibility of higher dimensions and non-orientable spaces. If the space is of higher dimensions than three, a left hand (LH) can be rotated through these extra dimensions to look like a right hand (RH). Similarly, if the embedding space is non-orientable, a LH can be translated in a path that turns it into a RH2. One can see this as an argument for externalism, since then the determination of left vs right is dependent on something external, say, human body, that is also embedded in the same space. As some philosophers point out[2, p4], this is a strong reason to reject internalist account in favor of externalism.
I, however, think this argument is misleading. For any n-Dimensional (nD) space, there will be totally asymmetric objects that have incongruent counterparts that are identical in an n+1D space. Hence postulating 4D space to reject internalism leads to an argument for infinite dimensions. On the other hand, if the higher dimensions are possible, one cannot say the differences are accounted by distance relations between two incongruent objects, since they also have to identify the dimensionality and orientation of the space they are working with. And if we are willing to include dimensionality in the discussion of asymmetry, it is better to absorb it into the meaning of “incongruent counterparts”; when someone asks whether objects p and q are incongruent counterparts the word incongruent has no meaning without specifying the possible rotations and translations – or the embedding space – of the two objects. As such, it does not seem that higher dimensions can provide support for either of the two relativist positions
Externalism (and relativism in general), on the other hand, have a more serious objection: the fall of parity. It is easier to discuss this issue by switching to a different way of looking at incongruity. Imagine us earthlings were to send information about earth to some unknown alien race3 and need to instruct them of the coordinate system (a right handed X-Y-Z axis) to use. We obviously cannot tell them its a RH coordinate system since there are no references that can define a RH, or put another way, there is no difference in how universe behaves when looking at its mirror image (called symmetry with respect to “Parity”). This is exactly the argument against internalism (and hence, for externalism), that there is no inherent difference between LH and RH. However, if there happens to be some law of nature4 that is asymmetric, we can use that as a reference to our RH coordinate system. This was the exact problem physicists (and philosophers, of course) had before 1950s. However, when Madam Wu et al[3] showed that the decay of 56Co is asymmetric in nature, our unknown-alien-communicators were rejoiced by the new-found ability to finally message the aliens that our hearts are on the left. Consequent development of physics have established that one of the four fundamental interactions in nature, weak interaction, is in fact fully asymmetric, and neutrinos (ν)5 exist only as left handed particles.
This revelation is a problem for externalism as it shows that one can refer to neutrinos (ν) to define handedness as they have intrinsic asymmetry by their mere existence. But is it so? One can further analyze the alien analogy to answer this question. When Alice, an unknown-alien-communicator stationed on earth tries to write the aliens that neutrino’s handedness correspond to our left hand, Bob, her colleague asks, ”how do they know its not an anti-neutrino (¯ν)6? Similar to νs that can only be left handed, ¯νs also can be only right handed. Hence, if the alien is made out of anti-particles, whose coupling to ¯νs is the same as our (particle-made) coupling to ν, he might try shake our hand with (what we call) his left hand; what an embarrassment at the first meeting! This question is in fact identical to the question of how to identify LH from RH; symmetry of combined operation of Parity(P) inversion and Charge conjugation(C) (or CP) (which turns a LH to a RH but also changes all its particles to antiparticles,) again leads us to believe the need for an external reference! This triumph of externalism, however, did not last long. When physicists in 1970s found out that CP too is not a symmetry in nature, it was possible to use those laws7 as a reference for the handedness. The externalist, however, can still appeal to the deeper symmetry in physics which combines CP with time reversal (T). Which is to say that a left handed ν traveling forward in time is identical to a right handed ¯νtraveling backwards in time, and argue this is still the same problem of identification. While this is clearly a valid approach, re-formulating incongruence to combine mirror images with particle-antiparticle conversion and time-reversal can be seen as less appealing. But luckily for the externalist, this is not the only avenue of argument.
As Pooley points out, externalism stands strong even if CPT symmetry is violated. The reason Alice can point to a ν and refer to it’s handedness is not because one ν is left handed, but because all the neutrinos are left handed. In fact, in a universe with only one ν, one cannot talk about a handedness or asymmetry in natural laws. The asymmetry materialize only with the appearance of the second and subsequent νs. To put this another way, the externalist can argue that when an internalist says the νs have a handedness, what he is really saying is that all the νs have the same handedness, and thus appealing to externalism. This can easily be extended to a universe in which even the CPT symmetry is violated, and seem to be a better approach than the previous one. However, there are arguments against this approach as well. As Pooley shows, arguing that handedness or νs appear only with the appearance a subsequent νs makes the theory non-local[2, p17]. While this does not have to be a problem, it is somewhat less pleasing given the laws of physics are formulated to have interactions that are only local. Even though this might, for some, weaken the appeal of externalism, it is clear that fall of parity is not a refutation of this point of view either.
While both the relational accounts of space have replies to problems risen with higher dimensions and parity violation, these arguments also somewhat weaken the appeal of these theories. On the other hand, with the acceptance of quantum field theory (QFT), one can easily argue that there essentially is no difference between space where there is a particle and where there is not. A point in space with a particle has a field with a particle number equaling to one, whereas a vacuum point has a field with a particle number zero. While it is arguable whether mathematical models of fields should be taken real or not, this approach clearly reduces all the particles and vacua to one quantum field (or a combination of multiple fields) with different particle numbers, thus averting the main problem for the absolutist: the postulation of existence of a separate entity for a particle. This point, however leads to a more involved discussion, and I think is out of the scope of this paper.
Incongruence of mirror images or asymmetric objects is an interesting problem that has implications on relativists vs substantivalist debate on space. While possibility of existence of higher dimensions seem to be a problem for internalism arguing for relativism, this is only a problem of how to pose the question; a problem of the definition of incongruence, and hence pose no real threat for the above view. Fall of parity, on the other hand, supports internalism while forcing the externalist to appeal to either a deeper symmetry of space, CPT, or to possible non-locality when analyzing objects and events related by mirror images. Given these problems and the success of quantum mechanics, I think it is possible to look at substantivalism in a more positive light, which unfortunately, should be a discussion for another paper.
1This is a brief introduction, see reference (Van Cleve) for full set of arguments.
2A non-orientable 3D space would have to be embedded in a higher dimensional Euclidean space, thus these might be actually similar arguments.
3In an unknown location in the universe – if we knew their location, we can tell them directions by referring to other stars
4A law of nature is seen to be some phenomena that is common both to us and the aliens
5ν is a weakly interacting particle that is the lightest known particle, is uncharged and is of paramount importance to the existence of life in our universe – disclaimer, I am a biased neutrino experimentalist.
6ν¯s are the antiparticles of , and is identical to them save for the fact that they couple to anti-particles the same way νs couple to particles. The real argument here is that one cannot independently identify a particle from an anti-particle without external reference to another (anti) particle. 7It is still the weak interaction, which violated CP symmetry at about level
References
[1] Van Cleve, J. Incongruent Counterparts and Higher Dimensions, §13 (From Metaphysics, The Big Questions) Oxford: Blackwell. 1998.
[2] Pooley, O. Handedness, Parity Violation and the Reality of Space, (http://philsci archive.pitt.edu/713/1/parity.pdf) 2002
[3] Wu, C. S, et al Experimental Test of Parity Conservation in Beta decay, Phyical Review 105(4):1413-1415 1957
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