Information and Energy

In classical physics with a particle ontology, the entire history of the universe could be encoded as the positions and momenta of the constituent particles at any given instant (timeslice). With deterministic dynamical laws this leads to the Laplacian idea of a block universe, in which everything, past and future, is implicit in the information (positions and momenta) at any given instant. No new information is ever created, and no information is ever destroyed. The conservation of information corresponds to the conservation of energy and momentum, as enforced by the physical laws. The particles are localized, and information on any given “origin” time slice can be conveyed from place to place only by means of particles. In fact, since the information consists entirely of the positions and momenta of particles, the particles themselves embody the localized pieces of information. (We don’t consider aggregates of particles, such as brains, with putative emergent level of information not reducible to the information of the particles.) There is no “telepathic” signaling or energyless transfer of information, since information can only be conveyed by the re-arrangement of energy i.e., of the particles.

Likewise with a wave ontology, from the standpoint of ordinary signal processing, the energy of a signal is the integral of the square of the amplitude, so every signal has positive energy. It can be shown that although the phase and even the group velocities of a wave can exceed the speed of light, the signal velocity (i.e., the speed at which information is conveyed) cannot exceed the speed of light. Indeed one of the foundational principles of special relativity is that no energy or information can be propagated faster than light. It is not just coincidental that these two things – energy and information – are both subject to the same limitation, because the conveyance of information requires a flow of energy (and every flow of energy conveys information).

In quantum physics, any energyless transfer of information would violate the Heisenberg uncertainty principle, since it would enable the transfer of information about positions and/or momenta of particles without any disturbance associated with energy flow (with the associated momentum).

Despite all this, some philosophers have argued that energyless signaling is possible, i.e., that it is possible to transfer information from one location to another, in flat 3+1 dimensional spacetime, without any energy flowing between those locations. In fact, they claim that energyless transfer of information is quite simple to achieve, and happens all the time. The example they cite can be expressed as follows: Suppose we are neighbors, and we’ve agreed that I will leave my porch light on until a burglar arrives, at which time I will turn the porch light off. In this situation, the philosopher claims that the absence of light energy constitutes the signal that a burglar has arrived, and hence this represents an energyless transfer of information. The reasoning is obviously fallacious, because the signal in this case is the trailing-edge step function that modulates the carrier wave (the porch light). If we idealize of the porch light as a continuous flow of energy, then the signal is just the transition of the step function as shown below.

To see clearly that the energy conveys the information, we need only note that the knowledge of the burglar’s arrival is not conveyed instantaneously, it is lagged by the time required for light to travel from house to house.

Now, the philosopher is undeterred by this, and attempts to salvage his claim by positing a square-wave carrier as depicted below.

He argues that the receiver can conclude that the burglar has arrived only after the amplitude has been zero for a half-period of the carrier wave. In other words, with this step-wave carrier there is a non-zero recognition time. Strictly speaking, every physical communication protocol entails a non-zero recognition time, since even the continuous porch light actually consists of a wave with a finite frequency, but the recognition delay is negligible in that case. Still, we can construct arbitrarily crude carrier waves with arbitrarily long recognition times. To emphasize this, the philosopher refines his argument to consider a carrier wave consisting of a sequence of equally spaced discrete energetic pulses, as shown below.

The pulses may be flashes of light or thrown pebbles or any other agreed discrete energetic actions, and the time interval between consecutive pulses is T. Suppose T equals 5 minutes, meaning that we’ve agreed to send a pulse every 5 minutes unless/until the burglar arrives. The philosopher contends that by not sending a pulse after the burglar arrives, we are sending a signal without energy. He says the neighbor can conclude the burglar has arrived at precisely 5 minutes (plus transit time) after the last energetic pulse, so the neighbor is acquiring new information at this instant, even though he is not receiving any energy at this instant.

Again, the philosopher’s reasoning is obviously fallacious. He fails to distinguish between information transfers and the drawing of inferences from previously received information combined with the passage of time with a previously-communicated protocol. First, from the standpoint of elementary signal processing theory, it is self-evident that the signal wave cannot be resolved any more precisely than the period of the carrier wave. If we are sending a pulse every 5 minutes and then decide to stop at some arbitrary point, the receiver can’t resolve our stopping time any more precisely than the 5 minute period between the receipt of the last pulse and the time of the first missing pulse. This is still just a trailing-edge step function, as in our first porch light illustration, except here the carrier wave is a low frequency sequence of discrete pulses.

But how, the philosopher asks, can we explain the fact that the receiver suddenly “learns” that the burglar has arrived at the time of the first missing pulse, when no energy is received at that time? Surely (the philosopher reasons) this shows that the receiver is receiving a (energyless) signal from his neighbor’s house at that moment. Needless to say (or so one would have thought), it shows no such thing. If the receiver had no sense of time, he could never conclude that the burglar had arrived. This makes it perfectly clear that there is no signal, energyless or otherwise, at the 5-minute mark. In order to perform the communication protocol, the receiver must not only be able to receive the energetic signals, he must also have his own internal clock. We can fully automate the process of triggering the alarm at the neighbor’s house by using the arriving pulses to reset a timer to zero. If the timer ever reaches 5 minutes and 1 second, the alarm is triggered. Thus the alarm is triggered by the receiver’s internal clock, initialized by each received pulse. Time plays a crucial role in this communication protocol.

To be explicit, when the neighbor receives a pulse at time t, he infers that the burglar had not arrived by the time t – L/v where L is the distance between houses and v is the speed of the signal. If the next expected signal is not received at time t + T, the receiver infers that the burglar arrived sometime between t – L/v and t + T – L/v. In other words, the last energetic pulse is the beginning of the uncertainty band on the burglar’s arrival time, and the size T of the uncertainty band is just the period between pulses, as previously communicated and agreed.

Occasionally one sees claims that, contrary to common wisdom and fundamental physical principles, someone has found an exotic way to transfer information without energy. For example, a 2015 paper (Jonsson, et al), while acknowledging that energyless transfer of information is not possible in flat 3+1 dimensional spacetime (which is the spacetime in which we live) argued that in hypothetical 2+1 dimensional spacetime there are off-shell effects for certain quantum interactions that would lead to afterglow, which (they argue) could be interpreted as energyless transfer of information. However, it’s well known that Hugyens’ Principle applies only in spacetimes with an odd number of space dimensions greater than 1, and this results in afterglow effects in spacetimes with an even number of space dimensions, rather than sharp on-shell propagation as in 3+1 spacetime. The novelty of the 2015 paper was to claim that this afterglow would represent energyless transfer of information when combined with quantum entanglement. But even this paper flatly refutes the philosopher’s claim that energyless transfer of information can exist in our flat 3+1 dimensional spacetime. (In curved spacetime the localization of energy itself becomes ambiguous, but our philosopher is not talking about curved spacetime.)

As an aside, it’s interesting to consider the microscopic attributes of an arbitrary electromagnetic wave, such as might be emitted from a radio antenna or, better, a single accelerating electron. Suppose the position of an electron is given by

This position could be plotted as shown below.

From quantum theory we know that electromagnetic radiation actually consists of photons whose energy is proportional to the frequency, but which frequency? Would the emitted photons have energy proportional to ω₁ or ω₂? We must expect photons with energies proportional to each of those frequencies. More generally, we could perform a Fourier analysis of an arbitrary wave form into its spectrum of frequencies, and we would expect a distribution of photons with individual energies corresponding to the various frequencies.

As an aside, note that if an electron were subject to constant proper acceleration for all time (past and future), we could not associate any frequency with that electron. If we cannot assign a frequency, how can it emit a quantum of energy? (The question of whether a uniformly accelerating electron radiates arises even in classical electrodynamics, which entails conflicts with causality, such as the pre-acceleration of an electron prior to the application of an electric field.) But any real electron must start and stop accelerating, so it has definite Fourier frequency components and can therefore radiate electromagnetic energy. However, this leads to the question of precisely when a photon is emitted, especially for the Fourier components of a square wave, when part of the waveform is flat, i.e., the charge is not accelerating. (Strictly speaking an ideal step function has no Fourier transform.)

With modern field theory and special relativity there is no such thing as “action at a distance”, where the word “action” refers to the physics definition, as in the principle of least (or stationary) action. (Interestingly, in a recent discussion, an individual who has written prolifically on the philosophy of physics remarked “You know, I never even considered taking the notion of "action at a distance" to have anything to do with the "action" defined in Newtonian physics!”) The traditional meaning of the word “locality” was that there is no instantaneous action at a distance, and with special relativity this was refined to no superluminal action. Hence energy and information can propagate away from a given event only within or on the forward light cone of that event. All evidence confirms that the principle of locality (defined as no superluminal action or propagation of energy or information) holds good without exception. Even quantum entanglement does not violate locality, providing perhaps the strongest confirmation of the validity of this principle. There is, however, a degree of correlation between coherent systems when the overall wave function cannot be factored, which is essentially the definition of quantum entanglement. The belief that spatially separated systems must be “factorable” in the sense that they can be treated independently with no violation of Bell inequalities, is called separability. Quantum theory entails the violation of separability, but not of locality.

Despite this, the foundational literature often contains claims that “quantum mechanics is non-local”. Two things need to be untangled to correct these claims. First, such claims are often based on non-relativistic quantum mechanics, which is inherently non-local, aside from any consideration of quantum entanglement. The constant c does not appear in the non-relativistic Schrodinger equation. In non-relativistic theory the Hamiltonian of a particle is p²/2m, so the particle can be accelerated to any speed. (Note that, if we assume the non-relativistic Schrodinger equation is true, then we have non-relativistic electromagnetism as well, i.e., not Lorentz invariant.) Thus, non-relativistic quantum mechanics is literally non-local, not because of quantum entanglement, but because it is non-relativistic, and hence there is no finite upper bound on the propagation speed of energy. (If a theory entails a finite upper bound on the propagation speed of energy, the theory must be relativistic and Lorentz invariant, because the postulate of a finite upper bound on speed - along with a few other commonly granted assumptions – implies Lorentz invariance.) Second, when people talk about proving that it’s impossible to send superluminal signals in non-relativistic quantum mechanics, what they really mean is that the quantum entanglement entailed by non-relativistic quantum mechanics (as well as by relativistic quantum mechanics) within the context of relativistic physics does not permit superluminal signaling. This is a very convoluted proposition, and it’s of interest only to dispel the idea that quantum entanglement, per se, enables superluminal signaling. But the important point is that non-relativistic quantum mechanics is inherently non-local in the sense that it permits superluminal signaling (albeit not using quantum entanglement), so it makes no sense to discuss whether the world is non-local in the context of a non-relativistic (and hence explicitly non-local) theory.

We need to talk in terms of relativistic quantum mechanics (quantum field theory), which is explicitly local, e.g., the Hamiltonian of a particle cannot be accelerated beyond the speed of light, and no superluminal communication is possible. Now, we still find the non-classical correlations between space-like separated measurements, due to the fact that the wave function of an entangled system cannot be factored (essentially the definition of entanglement). This signifies that the world is not separable, even though it is explicitly local.

Again, it is fundamentally misguided to consider non-relativistic quantum mechanics when discussing locality, i.e., the principle of no super-luminal action that is the foundation of special relativity. It’s true that quantum entanglement is already present in non-relativistic quantum mechanics, but the non-relativistic Schrodinger equation is not Lorentz invariant, so it simply is not suitable as the context for discussing locality. We need to focus on relativistic quantum mechanics, i.e., quantum field theory. Since quantum field theory is explicitly Lorentz invariant and entails no superluminal action or flow of energy or information, it explicitly satisfies the principle of locality – at least if the word locality has it usual meaning in physics.

How, then, can we account for the widespread claim that quantum entanglement proves that quantum theory is non-local? Recall that in the discussions of Bohr and Einstein the word locality was never used. Instead, they spoke of separability, i.e., the independence of spatially separate systems. Both agreed that quantum theory entailed a breakdown of the classical notions of separability, as was shown by the paper of Einstein, Podolsky, and Rosen (and even more clearly in Einstein’s later correspondence). The difference of opinion between Einstein and Bohr was that Einstein considered any failure of classical separability to be unacceptable, whereas Bohr was quite happy to accept it as a fundamental feature of the world. Einstein made a “slippery slope” argument, saying that if separability fails then science itself is impossible, because we can never separate anything from anything else in order to study individual things. This argument is not persuasive, because we don’t have complete failure of separability, but only enough failure to allow quantum entanglement, which is an experimental fact. The principle of locality (i.e., no superluminal propagation of energy or information) remains intact. Hence Einstein’s special relativity is not threatened, and the world exhibits a causal structure that is (at least somewhat) amenable to scientific study.

The use of the word “locality” (rather than separability) to refer to the principle that is violated by quantum entanglement seems to have originated with – or at least been popularized by – Bell’s papers, and thereby entered the literature of philosophers of science. But this is not so surprising if we realize that Bell was actually a neo-Lorentzian who believed (or strongly suspected) that special relativity actually was violated by quantum entanglement, and that something actually was being propagated faster than light. Of the four possible options for explaining the violations of Bell’s inequality in entangled systems, Bell’s clear preference was the third, which he described in his paper on Bertlemann’s socks as follows:

It may be that we have to admit that causal influences do go faster than light. The role of Lorentz invariance in the completed theory would then be very problematic. An ‘aether’ would be the cheapest solution. But the unobservability of this aether would be disturbing. So would the impossibility of sending ‘messages’ faster than light, which follows from ordinary relativistic quantum mechanics in so far as it is unambiguous and adequate for procedures we can actually perform…

He echoed this same idea consistently in his writings and interviews. In view of this, it isn’t surprising that he used the word locality (rather than separability), because he apparently really did believe (or suspect) that quantum entanglement involves a violation of locality, i.e., superluminal propagation of causal influences. We might say Bell made the same mistake that Einstein made by extrapolating from the manifest non-separability of quantum theory to the idea that quantum theory violates locality. Classically, the two concepts were essentially identical, but quantum theory teaches us that we must distinguish between locality (which is not violated) and separability (which is violated).

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