The Broken Symmetry of Time

The  Broken Symmetry of Time

 

A published version of this paper  can be found here:

http://link.aip.org/link/?APCPCS/1408/7/1

Ruth E. Kastner

Department of Philosophy, University of Maryland, College Park, MD 20742, USA

Abstract. This paper seeks to clarify features of time asymmetry in terms of symmetry breaking. It is observed that, in general, a contingent situation or event requires the breaking of an underlying symmetry. The distinction between the universal anisotropy of temporal processes and the irreversibility of certain physical processes is clarified. It is also proposed that the Transactional Interpretation of quantum mechanics offers an effective way to explain general thermodynamic asymmetry in terms of the time asymmetry of radiation, where prior such efforts have fallen short.

Keywords: time symmetry; quantum mechanics interpretations; Wheeler-Feynman theory; Transactional Interpretation

PACS: 04.20.Cv, 01.70.+w, 03.65.-w

1. What is symmetry breaking?

 

   In a nutshell, symmetry breaking is the selection of a particular event or situation from a set of possible alternative situations characterized by a symmetrical relationship. In more technical terms, symmetry breaking consists in the reduction of the symmetry group characterizing a given physical situation to a subgroup of the original group.  Symmetry breaking can be either forced or spontaneous: in the former, there is an asymmetric cause which picks out one from a set of solutions; in the latter, there is no asymmetric cause, yet somehow a particular solution is actualized.

In spontaneous symmetry breaking (SSB), the governing theory for the phenomenon under study specifies a symmetric situation, schematically illustrated in Figure 1.  A component of the theory (e.g., a field) undergoes a transformation in which a multiplicity of states or outcomes is possible, none of which can be ‘picked out’ by anything in the theory as the realized state or outcome.

Figure 1 Spontaneous symmetry breaking: a transformation of a theory component in which a multiplicity of states or outcomes is possible, none of which can be ‘picked out’ by anything in the theory as the realized state or outcome.

A specific example of this phenomenon occurs in the “Higgs Mechanism”[1] in what is termed the “Standard Model” of elementary particle theory. According to this widely accepted model pioneered by Steven Weinberg and Abdus Salam, the quanta of some force-carrying fields acquire a mass by way of a process in which the ground (vacuum) state of the field undergoes the kind of transformation conceptually depicted above. What was a single vacuum state of the field acquires what is termed  a “degeneracy”—that is, many possible ground states (in fact, an infinite number of them). The symmetry breaking occurs through what is called a “Mexican Hat” potential due to its shape.  The original ground state becomes unstable and corresponds to the crown of the ‘hat’; the infinite set of ground states are found all around the ring at the lowest point. The theory does not provide any way of deciding which of these many ground states is realized. But, according to the theory, the fact that the quanta in question have a nonzero mass indicates that one has been realized.

 

2. Curie’s Principle and Curie’s Extended Principle

The situation just described seems to run afoul of a philosophical doctrine[2]  termed “Curie’s Principle” in honor of Pierre Curie who championed it. (The principle is actually a version of Leibniz’ “Principle of Sufficient Reason” (PSR), which states that any event occurs for a reason or cause that specifies or determines that particular event, as opposed to some other event. The PSR implies that, absent such a reason or cause, the event in question will not occur.)

Curie’s Principle states that an asymmetric result (i.e., the choice of one outcome among many equally possible ones) requires an asymmetric cause. That is, it posits that there can be no sound basis for saying that one of the outcomes ‘just happens’; one must be able to point to a definite reason for that outcome (the reason being the asymmetric cause). This principle is illustrated by a humorous paradox, “Buridan’s Ass,” discussed by French philosopher Jean Buridan, in which a hungry donkey is placed between two equally distant, identical bundles of hay (see Figure 2). According to an implicit version of Curie’s Principle being satirized by Buridan[3], the donkey will starve to death because it has no reason to choose one pile of hay over the other. Of course, our ‘common sense’ tells us that the donkey will find a way to begin eating hay, even though one can provide no reason for it (hence the paradox). Similarly, in SSB, the field in question arrives in a particular ground state though no specific cause for that choice can be identified. If we take Curie’s Principle to be applicable to the above, then it appears that Nature simply violates the principle (as does a hungry donkey).[4]

Figure 2. A political cartoon (ca. 1900) satirizing U.S. Congress’ inability to choose between a canal through Panama or Nicaragua, by reference to Buridan’s Ass.

[Wiki Open Source; public domain]

There is another way of looking at this situation, described by Stewart and Golubitsky {1}. These authors point out that Nature seems to be replete with symmetries that are spontaneously ‘broken,’ similar to the way in which the symmetry of the vacuum state is broken by the Higgs et al. mechanism.  In general, a symmetrical system may, under certain circumstances, be capable of occupying any one of a set of symmetrically related states, with no particular state being privileged; thus the particular state in which it happens to be found is arbitrary. Stewart and Golubitsky suggest that this situation should be understood as conforming to an Extended Curie’s Principle (ECP), specifically: “physically realizable states of a symmetric system come in bunches, related to each other by symmetry”; or, alternatively, “a symmetric cause produces one from a symmetrically related set of effects.” (original italics; 1992, 60.) Technically, the ‘bunches’ are subgroups of the originally symmetry group which has been ‘broken’ by the dynamical situation under consideration. ECP amounts to   a weakening of Curie’s Principle.

As noted by Stewart and Golubitsky, a famous illustration of symmetry breaking appears in the iconic 1957 photo of the splash of a milk droplet by high-speed photography pioneer Harold Edgerton (see link indicated for “Figure 3”). The authors point out that the pool of milk and the droplet both have circular symmetry, but the ‘crown’ shape of the splash does not—it has the lesser symmetry of a 24-sided polygon. This happens because the ring of milk that rises in the splash reaches an unstable point—a point where the sheet of liquid cannot become any thinner—and ‘buckles’ into discrete clumps (the laws of fluid dynamics predict that there are 24 clumps).  But the locations of the clumps are arbitrary; that is, the clump appearing just beneath the white droplet above the crown could just as well have been a few degrees to the left (with all the other clumps being shifted by the same amount). An infinite number of such crowns are possible, but only one of them is realized in any particular splash.

FIGURE 3.  Harold Edgerton’s iconic milk droplet splash photo.

MIT© 2011 Massachusetts Institute of Technology, Courtesy of MIT Museum

Thus, the authors point out that, while the mathematics describing a particular situation may provide for a large, even infinite, number of possible states for a system to occupy, in the actual world only one of these states can be realized. They put it this way:

“A buckling sphere can’t buckle into two shapes at the same time. So, while the full potentiality of possible states retains complete symmetry, what we observe seems to break it. A coin has two symmetrically related sides, but when you toss it it has to end up either heads or tails: not both. Flipping the coin breaks its flip symmetry: the actual breaks the symmetry of the potential.” (1992, 60.)

The last sentence has been italicized because it expresses a deep and important principle: mathematical descriptions of nature, with their high degree of symmetry, in general describe a world of possibilities rather than a specific state of affairs. Nevertheless, the astute reader may well raise the following question: but isn’t it the case that, in the classical domain, we can always find some external influence, however small, that caused the system to end up in one particular state as opposed to some other possible state? This would seem to apply, for example, in classical chaotic systems such the double pendulum (See Figure 4). For large initial momentum, such a system’s set of possible trajectories encounter ‘bifurcation points’ (essentially ‘forks in the road’) in which a specific choice of trajectory is sensitive to perturbations down to the Planck scale (i.e., random quantum fluctuations).

 

Figure 4. A double pendulum, whose classically-described motion

encounters bifurcation points.

The authors address this, at least in part, as follows:

“we said that mathematically the laws that apply to symmetric systems can sometimes predict not just a single effect, but a whole set of symmetrically related effects. However, Mother Nature has to choose which of those effects she wants to implement.

How does she choose?

The answer seems to be: imperfections. Nature is never perfectly symmetric. Nature’s circles always have tiny dents and bumps. There are always tiny fluctuations, such as the thermal vibration of molecules. These tiny imperfections load nature’s dice in favor of one or the other of the set of possible effects that the mathematics of perfect symmetry considers to be equally possible.” (1992, 15)

Thus, the apparent answer of the authors to the question of what causes the system to end up in a particular state is: quantum fluctuations. That is, the cause is found outside the mathematical formulation of the set of possible solutions for the classical system (or, in the case of certain chaotic system such as the double pendulum above, by following the classical account into the quantum domain in which its deterministic aspect breaks down). It appears that, strictly speaking, when considering symmetry breaking in the classical domain, one could always point to some external cause of this type, even if only a random quantum fluctuation. So when the authors say that “the actual breaks the symmetry of the potential,” they are not yet describing the quantum domain. Instead, they are describing the realization of a particular classical state from an idealized, abstract set of equally possible states, where the realization can be attributed to the existence of a quantum domain that can ‘precipitate’ a particular classical state by way of random quantum fluctuations. One can therefore point to the fluctuation precipitating the specific outcome as the ‘asymmetrical cause’ required by Curie’s Principle.

If we return to the case of spontaneous symmetry breaking in the Standard Model, clearly we are dealing with symmetry breaking in a purely quantum context: the system comprises the vacuum and the Higgs field, purely quantum entities. If we want to try to follow the same procedure and to seek a specific cause — however fleeting and random — for the choice of one of the infinite set of possible vacuum states, we either have to suppose that it also stems from fundamentally indeterministic quantum fluctuations of the vacuum, or postulate fluctuations in some deeper realm that lies outside any current theory. The point is still that “the actual breaks the symmetry of the potential,” however this is accomplished. The only alternative is to postulate that SSB in the Standard Model requires a ‘many worlds’ interpretation, in which SSB gives rise to many possible worlds, each with a different vacuum state. But this is certainly not the usual approach, which simply assumes that the actual universe corresponds to one particular vacuum state.

The authors of “Fearful Symmetry” further note that instances of symmetry breaking give rise to concrete structures that often seem to reflect design or intent. An interesting example is found in crop circles. The authors point out that an unblemished field of corn has a very high degree of symmetry: translation, reflection, and rotational symmetries. If symmetry is broken at a point—say by a falling object, such as a hailstone—the effect will ripple out radially to create a circle. A particular rotational center has been chosen from among an infinite number of equally ‘eligible’ ones, and concrete structure is born. The structure must obey the underlying rotational symmetry and it is also constrained by the nature of the objects comprising it. In the case of the crop circle, the physical properties of the cornstalks, together with the energy of the falling object or other precipitating event, will dictate the radius of the circle.

The point here is that the appearance of organized, symmetrical structures in the empirical world should not be taken as an argument against asymmetry but rather as an argument in favor of the need for asymmetry; that is, one requires lesser symmetry than the underlying theory governing possibilities rather than actualities, if one is to have an empirical world of experience with concrete objects, systems, and events.

 

3. Time and symmetry breaking

Before considering time symmetry and its breaking, we need to distinguish two features of time symmetry that are often conflated: (1) the anisotropy of all temporal processes and (2) the irreversibility of certain temporal processes. Feature (1) is simply the observation that temporal events constitute an ordered sequence that proceeds, like a set of movie frames, in only one direction (i.e., monodirectionally as the index t increases or decreases). This is qualitatively different from the case of spatial events which (in one spatial dimension x) can be bidirectional or (in three spatial directions) can be omnidirectional, like an expanding spherical wave.

Feature (2) is the observation, familiar from thermodynamics,  that if we ‘run the movie backwards’ most macroscopic processes (such as cream mixing into coffee)  look physically unreasonable; while similar time-reversals of microscopic processes (such as small numbers of air molecules in a box) look reasonable. The former processes are termed ‘irreversible’ and the latter ‘reversible.’ But it is important to keep in mind that there is a distinction between (2) the observation that ‘running the movie backwards’ can produce unrealistic phenomena, and (1) the observation that there is a ‘movie’ in the first place. It is to (1) that we now direct our attention, since it is more fundamental.

What kind of world would we have if temporal processes were just as isotropic as spatial processes? On reflection, we can’t even conceive of such a world in any empirical sense; for in order to perceive anything, we can only perceive it as a sequence of events. One might call this a ‘transcendental argument’ for the anisotropy of time; the empirical world of our experience is simply not recovered for isotropic temporal processes, whatever that might mean. In other words, in order for us to experience what we know we experience, events must proceed monodirectionally in temporal index t.  By convention, we designate that direction by increasing values of t. Thus the ‘breaking’ of time symmetry (i.e., time isotropy) seems required a priori, simply to account for the possibility of empirical experience. In what follows we examine the breaking of time symmetry as also contingent on energy propagation.

4. Time symmetric theories

While we can’t experience isotropic temporal processes, there nevertheless are temporally isotropic theories.  These are conventionally called “time symmetric” theories, so let’s revert to that usage, keeping in mind that ‘time symmetry’ here really means ‘time isotropy.’ An example is the Wheeler-Feynman theory of classical electrodynamics {2} (‘WFED’), which uses a time-symmetric field together with time-asymmetric cosmological boundary conditions to recover the apparent time-asymmetric fields of standard classical electrodynamics (CED).

One might wonder why we should entertain a time-symmetric theory, which seems counterintuitive. The motivation for doing so is that our unidirectional temporal experience is evidence of a broken symmetry. Symmetries are broken by way of boundary conditions, such as those arising from the constraints on the milk droplet as it hits the pool, even if they don’t determine the ultimate position of the milk droplet coronet. Just as we don’t apply boundary conditions until the droplet hits the pool, the most general approach is to refrain from imposing boundary conditions until the underlying law is confronted with contingent features of our universe. The methodological advantage of this approach is that it avoids imposing possibly ad hoc explanations and conditions which may not actually hold in our universe; instead, it allows the theory itself to tell us what is required for the contingent asymmetries that we experience.

It must also be emphasized that, in finding general solutions for wave equations describing fields, specific boundary conditions must occur in order for energy to be propagated by way of those fields. That is, if one assumes a point source for the field (this corresponds to the inhomogeneous field equation), the solution is singular for real energies. One cannot obtain a physically meaningful solution (Green’s function, also termed a ‘propagator’) without analytic continuation of the frequency (energy) coordinate and choice of a contour of integration; the latter corresponds to choice of boundary conditions. This suggests that fields arising from sources cannot actually propagate energy unless specific boundary conditions (corresponding to a choice of integration contour) exist.[5]

One such propagator is the ‘retarded’ propagator which allows for propagation of positive energies only in the forward time direction (into the future). This solution is the one used in classical electromagnetic theory; the advanced solution which allows negative energy to propagate into the future (or positive energy to propagate into the past) is simply dropped as ‘unphysical.’   However, this approach does not account for the loss of energy (‘radiative damping’) by an emitting particle, and an ad hoc additional free field must be assumed (see below).

Wheeler and Feynman (WF) decided to explore a time-symmetric approach because they were not satisfied with the standing method of dealing with radiative damping.  Dirac {3}{ had proposed that damping can be explained by a free field (that is, a field not attributed to any source) in addition to the basic retarded (unidirectional, positive t direction) propagation by the charge.  While this seemed to account for the observed energy loss, WF were dissatisfied by its ad hoc character. They proposed instead that the basic propagation due to the charge was time-symmetric,  where the time symmetric propagator is simply the sum of half the retarded and half the advanced propagators. WF proposed that other charges (absorbers) responded to that initial time-symmetric field by emitting their time-symmetric field out of phase with the stimulating field.  If the universe is a ‘light tight box’—if there are sufficiently many absorbers for each emitter – the collective response of the absorbing particles turns out to provide, at the location of the emitting charge, the apparent ‘free field’ needed to account for loss of energy by the radiating charge. It also provides for cancellation of the retarded field beyond the absorbers (which is why they absorb), and of advanced propagation (of positive energy into the past) due to the emitter (so no residual advanced effects remain). Thus the asymmetric boundary condition of the preponderance of absorbers vs. emitters provides for the apparent time-asymmetry of radiation, as well as a natural (non-ad hoc) explanation for the absorption of energy and of radiative damping).

The WF theory thus describes radiation of energy as a direct (time-symmetric) interaction between sources (sinks), and the emitting particle is taken as not interacting at all with its own emitted field (where the latter process is commonly referred to as ‘self-interaction’).  Such theories are called “direct action” (DA) theories. However, it later became evident that some form of self-interaction was needed to account for certain relativistic effects (such as the Lamb shift {4}). Davies {5} introduced this feature into a quantum relativistic extension of the Wheeler Feynman theory.

It remains a matter for further investigation as to whether DA-type theories are empirically equivalent to standard quantum field theories (QFT). However, to date there is no conclusive evidence or argument that DA theories are not empirically equivalent to QFT, and there is much to recommend them in methodological terms, as argued above. Some researchers dislike DA theories because they are generally impractical for doing computations, since they depend explicitly on the actual boundary conditions of the generating sources and sinks (for which the technical term is ‘currents’) both in the future and in the past; there is no independent field with degrees of freedom of its own that can be quantized (i.e., treated as harmonic oscillators with discrete states of excitation). In contrast, standard field theories use quantized fields as independent entities and therefore do not need to explicitly refer to the currents that generate them. While standard quantum field theories are therefore much better computational tools, that pragmatic fact does not rule out the distinct possibility that Nature actually uses direct action in the universe , of which we can only study a small portion in any given computation.

 

5.  Boundary conditions and the arrow of time

As noted above, assuming only retarded field propagation does not allow for radiative damping; an ad hoc free field must be imposed. A more natural and general approach takes the underlying propagation as time symmetric (isotropic) and seeks to discover what actual boundary conditions must exist in order for energy to be transferred from one place to another in accordance with empirical observation. This turns out to be the condition that the universe is a ‘light-tight box’; i.e., fields do not propagate to infinity.

It has been observed in the past (e.g. Ritz 1909, as  quoted in  Zeh {7}) that one could  relate the apparent asymmetry of electromagnetic radiation to the thermodynamic ‘arrow’ – i.e.,  the preponderance of irreversible physical processes (such as the mixing of coffee and milk discussed above). However, those approaches simply omitted the advanced electromagnetic wave solution a priori and assumed that the asymmetry of the retarded radiation solution alone could be extended to the general thermodynamic asymmetry.[6] This was problematic because the latter applies to neutral particles as well, and would therefore seem to have nothing to do with electromagnetic fields. If, in contrast to the traditional rejection of advanced solutions, we suppose that the underlying laws are truly time-symmetric, then the apparent asymmetry of radiation (i.e., retarded only) is due to the asymmetry of boundary conditions involving the distribution of charges. But the question remains: how can this approach to electrodynamics be extended to neutral particles? A possible answer may be found in a time-symmetric interpretation of quantum mechanics based on the WF approach: the Transactional Interpretation.

6. The Transactional Interpretation

John G. Cramer introduced the Transactional Interpretation (TI) of quantum mechanics in the 1980s {8}. He saw TI as a natural generalization of the Wheeler-Feynman approach that would serve to explain the Born Rule yielding the probabilities for results of measurements; he also saw advanced solutions as manifest in such ubiquitous features of the Hilbert Space quantum formalism  as inner products.

TI proposes that the standard quantum state (‘ket’) corresponds to the retarded electromagnetic solution, and the adjoint quantum state (‘bra’) corresponds to the advanced electromagnetic wave solution.  It proposes that whenever a source (in this case a source (current) for the quantum field corresponding to the quantum under study) emits the field solution described by a ‘ket’ it also emits the field solution described by a ‘bra’, and that other currents respond to the emitted field exactly as the ‘absorber’ in the WF formalism: i.e., out of phase such that the advanced field of the emitter and the retarded field beyond the absorber is cancelled.  As in the WF theory, transfer of energy takes place from the emitter to the absorber. The difference between TI and the WF picture is of course that the former is a quantum process in which energy is transferred as discrete packets rather than continuously. This implies that, even though many ‘absorbing’ currents are involved in the necessary cancellation and reinforcement of the requisite fields, only one can be chosen for the ultimate transfer of energy. Such potential transactions (i,e. potential transfers of energy from the emitter to each of the participating absorbers) are called ‘incipient transactions’ and the one chosen out of the set of incipient transactions is called an ‘actualized transaction.’

If the preceding situation sounds familiar, it should; it is analogous to spontaneous symmetry breaking as in the case of the milk droplet and the Higgs mechanism. The only difference is that in TI, some incipient transactions may be more probable than others. In fact, their probabilities are precisely given by the Born Rule (as demonstrated in Cramer 1986). It is suggested that this can be viewed as a kind of ‘weighted symmetry breaking’ in that clearly not all transactions can be realized, but some are more probable than others.

Now, recall that TI applies to all quantum fields, including neutral ones. While thermodynamic irreversibility is usually thought of as applying to macroscopic (classical) systems, it must be kept in mind that the quantum level is the more fundamental one which must underlie all classical phenomena.  The transactional interpretation of quantum theory tells us that all transfers of energy (and other conserved physical quantities) take place due to the interaction of an emitted field with ‘absorbers’ conforming to the boundary conditions needed for cancellation of residual advanced and retarded fields—where the latter are all quantum fields, not just charged ones. It can thus account for general thermodynamic irreversibility in the same terms as the asymmetry of radiation, by attributing both to the breaking of symmetry of an underlying theory.

The picture that emerges is the following:  symmetrical physical laws describe potentialities, not actualities. In order to have actual events in an actual world, the symmetries of those laws must be broken by the imposition of constraints in the form of boundary conditions. Such boundary conditions may not always specify which actual event or form will exist – often that specific event will arise from spontaneous symmetry breaking — but they serve to precipitate that actuality.  In the case of Edgerton’s milk droplet coronet structure, the precipitating boundary condition is the physical limit of the thinning of the milk surface in the rising splash. In the case of energy propagation, the precipitating boundary condition is the preponderance of absorbers compared to emitters; and further, in the quantum case, the restriction of energy propagation to discrete quanta. Neither the milk droplet coronet structure nor the transfer of energy would take place in the absence of the relevant boundary conditions.

Since the direction of positive energy transfer dictates the direction of change (the emitter loses energy and the absorber gains energy), and time is precisely the domain of change (or at least the construct we use to record our experience of change), it is the broken symmetry with respect to energy propagation that establishes the directionality or anisotropy of  time. [7]  The reason for the ‘arrow of time’ is that the symmetry of physical law must be broken: ‘the actual breaks the symmetry of the potential.’

It is often viewed as a mystery that there are irreversible physical processes and that radiation diverges toward the future. The view presented herein is that, on the contrary, it would be more surprising if physical processes were reversible, because along with that reversibility we would have time-symmetric (isotropic) processes, which would fail to transfer energy, preclude change, and therefore render the whole notion of time meaningless.

7. Conclusion

It has been argued that the ‘arrow of time’ is a result of symmetry breaking of the physical laws governing energy propagation. The same boundary conditions necessary for propagation of energy in a time-symmetric theory may serve to explain thermodynamic irreversibility when that theory is extended to the quantum domain which underlies all macroscopic processes. Thus the most economical and natural explanation of both aspects of the ‘arrow of time’ is that basic physical laws are symmetrical with respect to both space and time but describe only potentialities, and that actual events and processes arise because of symmetry breaking due to contingent boundary conditions.

Acknowledgements. The author gratefully acknowledges an invitation from Daniel Sheehan to deliver this paper and participate in the 2011 AAAS Symposium on Retrocausality in Quantum Mechanics

 

 

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