Interpreting Quantum Mechanics

In my last post, I discussed the mathematical intuition underlying Quantum Mechanics, which is the idea that particle configurations with probability amplitudes can cancel out when combined. That is because amplitudes can be positive, negative or complex numbers, not just positive numbers as classical probabilities are.

I also noted that when a photon arrives at the back screen in the double-slit experiment, we only see one of the slit detectors activated, which corresponds to one of the configurations. But how do we account for the configuration where the other detector was activated?

There are several interpretations, all disturbing, so therefore best described by reference to superheroes.

Copenhagen Interpretation: You, the conscious observer, create reality. If you look at the back screen, the photon will be a wave. If you look at the slits, the photon will instead be a particle. You don't feel like you're in a superposition, so you're an exception to the laws that those little particles follow.




De Broglie–Bohm (or pilot-wave) Interpretation: The photon surfs on a wave which carries it through one of the slits to a location on the back screen where destructive interference doesn't occur. Added bonus: Entangled particles can communicate faster than the speed of light in violation of special relativity.

Many-Worlds Interpretation: You observe one of the photons going through one slit while your twin observes the other photon going through the other slit. You are either observing a superposition on the back screen or participating in a branch of one when you become entangled with one of the photons. Also the entire universe is in a superposition. Things just look classical because you have to be standing in one configuration or another.





Instrumentalist Interpretation: Yada yada yada ... who cares? You can build really amazing stuff using quantum mechanics! Also affectionately called, "shut up and calculate!".

So there we have it. The Copenhagen Interpretation posits an observer-dependent reality and a mysterious wave function collapse. The Bohmian Interpretation requires information to travel faster than light.^[1] The Many-Worlds Interpretation causes incredulous stares. And, finally, the Instrumentalist Interpretation isn't an interpretation at all.^[2]

Since each interpretation uses the same mathematical formalism, is there any reason to prefer one to another on philosophical grounds? I think there is.

The famous Schrodinger's Cat thought experiment vividly demonstrates the logic of quantum behavior in terms of familiar, everyday things. The upshot is that the cat is in a superposition of being both dead and alive until we look in the box. But what does that mean? And why don't we normally observe such things?

To take the second question first, we actually can observe superpositions involving objects that are (just barely) visible to the naked eye. For example, a recent experiment demonstrated interference effects for the superposition of a tiny tuning fork vibrating and not vibrating. This is analogous to the double-slit experiment where photon amplitude flows through both slits and we observe an interference pattern on the back screen.

So what does this mean? It means that we have observed the effects of a single amplitude, and it is the sum of the amplitude for a vibrating tuning fork and the amplitude for a non-vibrating tuning fork.

How can this observation be explained in a coherent way? The idea that there is a single tuning fork that is both vibrating and not vibrating is a contradiction, so that fails. The idea that there is a single tuning fork that is either vibrating or not vibrating also fails, since possibilities can't cause interference effects. That leaves the idea that there is one tuning fork that vibrates and a second one that does not and that we are observing the combined effects of both.

If there are two tuning forks, what would explain the observation of a single tuning fork (that is either vibrating or not) when you try to detect the vibration? There are paths that entangle you with the vibrating tuning fork and paths that entangle you with the non-vibrating tuning fork and amplitude flows along both paths. Becoming entangled with one of the tuning forks destroys the interference pattern from your point-of-view (that is, you can no longer observe the effects of the other tuning fork because there are no amplitude paths from here to there).^[3]

The Many-World's Interpretation, despite its startling implications, seems to me to be the most coherent interpretation of both our everyday and quantum observations. That, of course, doesn't mean that it is true - that is ultimately an empirical question^[4]. However it is an intuitive and natural framework for conceptualizing our observed experience.

--

[1] See Bells Theorem which rules out local hidden variables.

[2] The pragmatic and reasonable version of the Instrumentalist Interpretation is that the math works, whether or not we know what it means. In stronger versions, it asserts that there is no explanation to be had and that the math doesn't mean anything.

[3] This process is called decoherence and is only reversible in practice in microscopic environments where interactions with air and apparatus molecules can be controlled (it's easy to break an egg but difficult to put it back together again). So observing things in a single (or decohered) state is the everyday situation that corresponds to our classical intuitions. The difficult challenge for quantum computing just is how to reliably maintain quantum bits in a superposition of 1 and 0 values.

[4] Scott Aaronson has a great post explaining that while the Many-Worlds Interpretation is the obvious, straightforward reading of quantum mechanics, it is also provisional in a way that heliocentrism (as opposed to geocentrism) isn't. You could, in principle, fly a spaceship above the plane of the solar system and see the Earth and the other planets circling the sun. However you can't similarly travel to another world branch to meet your twin.

Visualizing Quantum Mechanics

Diagram 1: Wave-like interference pattern

In 1803, polymath Thomas Young performed the famous double-slit experiment demonstrating that light exhibits wave-like behavior. When light is shone on a plate pierced by two parallel slits, an interference pattern appears on the screen behind the plate. No light appears at those locations where the wave crests and troughs combine and cancel out, as shown in Diagram 1. Also, the light is most intense at the center of the screen where the waves combine constructively.

In more recent double-slit experiments it has been shown that the same interference pattern emerges even when light is emitted one photon at a time! How can this be? Surely each single photon must go through either one slit or the other, but then why wouldn't that produce the two-clumps pattern^[1] shown in Diagram 2 below? Perhaps an experiment could be done to observe whether the photon really does pass through one slit or the other.


Diagram 2: Particle-like versus wave-like behavior

So when detectors are placed at the slits, each emitted photon is detected passing through just one of the slits, just as we would expect, and not through both slits as a wave would. However the interference pattern then disappears!

Strangely, the photon exhibits particle-like behavior when we detect which slit it passes through but exhibits wave-like behavior when we don't try to detect which slit it passes through. Does the act of observation determine the behavior of the photon?^[2]

These observations and questions highlight the generally-perceived mystery around quantum mechanics. The purpose of this post is to describe a way to visually conceptualize what is going on in the double-slit experiment using the correct mathematical intuition. This can help us to think more clearly about these questions. So let's get started!

Consider a photon that is emitted toward the plate in Diagram 2 above. Let's also consider a single location on the back screen, labeled B0, where destructive interference would occur. There are two paths from the photon emitter to B0, one through each slit, labeled S1 and S2. A distinct state that a photon and experimental apparatus can be in (such as their positions) is called a configuration^[3].

A configuration has a value associated with it called an amplitude^[4]. An amplitude is expressed as a complex number in the form (a + bi) and can be visualized as an arrow that can point in any compass direction. Amplitude flows from prior configurations to subsequent configurations. For each configuration, the incoming amplitudes are summed. The amplitude is also transformed by rules, such as when a photon moves, changes direction through a slit or activates a detector.

Starting with the initial configuration (and amplitude) for our double-slit thought experiment, we can transition to subsequent configurations, follow the rules that transform amplitude, and see where we end up.

The configurations (including the amplitude transformation rules)^[5] are:

1. Initial configuration: (-1 - i) [arrow pointing south-west]
2. A photon goes from the emitter to S1: multiply by -1 = (1 + i) [north-east]
3. A photon goes from S1 to B0: multiply by -i = (1 - i) [south-east]
4. A photon goes from the emitter to S2: multiply by (0.5 + 0.5i) = (0 - i) [south]
5. A photon goes from S2 to B0: multiply by (-1 - i) = (-1 + i) [north-west]
6. A photon arrives at B0: (0 + 0i) [no arrow]

The total amplitude flowing toward B0 is the sum of the individual amplitudes flowing toward B0 (underlined), which is (1 - i) + (-1 + i) = (0 + 0i). The probability of a photon arriving at B0 is the squared modulus of the amplitude (a² + b²)^[6], which is 0² + 0² = 0%. Therefore no photon will arrive at B0, due to the individual amplitudes canceling each other out (i.e., the equal-length arrows pointing in opposite directions). This corresponds to the wave-like behavior that is observed when a series of emitted photons create an interference pattern.^[7]

Now consider a second experiment where photon detectors are added at slits S1 and S2, labeled D1 and D2 respectively. They will turn from off to on if they detect a photon passing through their slit. The rules are the same as for the first experiment. The configurations are:

1. Initial configuration: (-1 - i) [arrow pointing south-west]
2. A photon goes from the emitter to S1 and D1 is off and D2 is off: (1 + i) [north-east]
3. A photon goes from S1 to B0 and D1 is on and D2 is off: (1 - i) [south-east]
4. A photon goes from the emitter to S2 and D1 is off and D2 is off: (0 - i) [south]
5. A photon goes from S2 to B0 and D1 is off and D2 is on: (-1 + i) [north-west]
6. A photon arrives at B0 and D1 is on and D2 is off: (1 - i) [south-east]
7. A photon arrives at B0 and D1 is off and D2 is on: (-1 + i) [north-west]

In this experiment, the individual amplitudes flowing toward B0 (underlined) are flowing to two distinct configurations (since there can be no single configuration where D1 is both on and off), so the individual amplitudes are not summed. There is now a positive probability that the photon arrives at B0, with an equal probability of the photon being detected at either slit (the squared modulus of the amplitude for each final configuration is 2, so the ratio is 2:2). This corresponds to the classic particle-like behavior that is observed when the photon is detected going through one of the slits.^[8]

So complex addition of destination configuration amplitudes is the mathematical basis for our observations in the double-slit experiment. The classical intuition is that adding more paths to a destination makes it more likely to reach the destination. The quantum intuition is that adding more paths to a destination can make the destination unreachable since paths can destructively interfere.

This still leaves one more puzzling question. When a photon arrives at B0, we only see one of the detectors activated, which corresponds to one of the configurations. But how do we account for the configuration where the other detector was activated? I will leave this question for a future post.

The ideas presented here were inspired by Eliezer Yudkowsky's post on configurations and amplitude from his series on quantum physics. For my earlier posts on visualizing complex numbers, see Seeing complex numbers and Visualizing Euler's Identity.

--

[1] To visualize photons exhibiting particle-like behavior, imagine someone with a gun firing bullets at the plate. For the bullets that pass through the slits, one clump of bullets would accumulate behind the first slit and a second clump of bullets would accumulate behind the second slit.

[2] That is, how should these counter-intuitive observations be interpreted? For one example, theoretical physicist John Archibald Wheeler once commented, "Actually, quantum phenomena are neither waves nor particles but are intrinsically undefined until the moment they are measured. In a sense, the British philosopher Bishop Berkeley was right when he asserted two centuries ago 'to be is to be perceived.'" - Scientific American, July 1992, p. 75

[3] A configuration (quantum state) is a distinct state that the universe is in at a point in time. In reality, it includes all the particles in the universe and all the particles that the emitter, plate, detectors and human observers consist of. What distinguishes one configuration from another is that at least one particle has a different property or position. When two configurations interfere, they combine to form a single configuration - an instance of the superposition principle.


Diagram 3: (a) Rotation to sine wave (b) 180^o phase shift

[4] The amplitude of a wave is the magnitude from rest to crest and is a real number - see Diagram 3. The configuration amplitude, which is the sense used here (and is elsewhere termed a probability amplitude), is a complex number (or phase vector) which additionally encapsulates a phase angle and can be visually represented as a radial arrow that points in any compass direction. For example, (0 + i) represents an anti-clockwise rotation of 90^o from 1 on the real number line and corresponds to an arrow pointing north (the wave peak). Combining two similar waves that are phase-shifted by 180^o (equal-length arrows pointing in opposite directions) results in wave cancellation (destructive interference).

[5] The initial configuration amplitude and the rules in the thought experiment are hypothetical, but serve to demonstrate the key conceptual point of amplitude interference. For the first photon path to B0, (-1 - i) * -1 * -i = (1 - i). For the second photon path to B0, (-1 - i) * (0.5 + 0.5i) * (-1 -i) = (-1 + i). These two final amplitudes have the same magnitude but are 180^o out of phase and therefore cancel out when in a superposition.

[6] This is known as the Born rule.

[7] We can also consider the center of the back screen where the light is most intense. In this case, two incoming configuration amplitudes with the same phase angle are constructively interfering, thus summing their magnitudes. For example, (1 - i) + (1 - i) = (2 - 2i) which corresponds to an arrow sqrt(2² + 2²) units in length pointing south-east.

[8] The detector (with an on or off state) could equally be replaced by a rock (that is or is not perturbed by a photon) and the computational logic would be the same.

The moral law

In my last post I discussed the solution to the Euthyphro Dilemma. It so happens that a parallel dilemma can be constructed for ourselves. Do we approve of an action because it is good or is an action good because we approve of it?

For morality to be objective it must be based on something other than subjective opinion. In the case of humans, that something is human nature [1]. What distinguishes us from the rest of the animal kingdom is our capability for rational thought. Unlike other animals, which automatically act on instinct to survive, we face a choice - to live or to die. To live (and flourish) is the value that bridges the gap from the "is" of human nature to the "ought" of morality. [2]

An analogy may be helpful here. Medicine is a normative science based on physiology which presupposes the value of health. Ignoring facts about physiology leads to poor health. Similarly, morality is based on human nature and presupposes the value of life and well-being. Ignoring facts about human nature leads to misery for ourselves and society.

Normative terms such as "moral", "virtuous", "good", "bad", "evil" and so on derive their meaning from the choices we make in ordinary, everyday contexts (with well-being as the standard of value). There is substantial agreement across cultures about the content of morality, as evidenced by the widespread injunctions against murder, violence, theft and so on, and in commonly found maxims such as the Golden Rule. [3] Differences that we do find can be recognized as one of degree rather than as being moralities of a radically different kind. [4]

--

[1] For theists, human nature is usually understood to reflect God's nature which allows the moral law to be discoverable. Otherwise, as C.S.Lewis says, "If we once admit that what God means by "goodness" is sheerly different from what we judge to be good, there is no difference left between pure religion and devil worship." (From the "The Poison of Subjectivism" in "Christian Reflections".)

[2] In Aristotelian terms, humans are rational animals and human flourishing (eudaimonia) is the final cause (telos) that grounds moral actions. We can see precursors to value, purpose and choice in an animal's survival instinct. But value, purpose and choice are intentional terms that derive their literal meaning from their use in everyday human contexts.

[3] See the appendix of "The Abolition of Man" where C.S.Lewis presents textual evidence of a universal moral law across modern and ancient cultures.

[4] C.S.Lewis discusses this point in "The Poison of Subjectivism". He notes that the Nietzschean ethic is innovative not because it grounds a different kind of morality but because it rejects objective morality altogether. That is, Nietzsche accepts the subjective horn of the dilemma.

Euthyphro's Dilemma

In "The Poison of Subjectivism", C.S.Lewis asks, "But how is the relation between God and the moral law to be represented? To say that the moral law is God's law is no final solution. Are these things right because God commands them or does God command them because they are right?" [1]

This is the modern version of the dilemma first posed by Socrates in Plato's "Euthyphro". Lewis explains why he is unable to accept either horn of the dilemma.

"If the first, if good is to be defined as what God commands, then the goodness of God Himself is emptied of meaning and the commands of an omnipotent fiend would have the same claim on us as those of the "righteous Lord." If the second, then we seem to be admitting a cosmic dyarchy, or even making God himself the mere executor of a law somehow external and antecedent to His own being. Both views are intolerable."

Be that as it may, rejecting the dilemma is not a valid option here. In essence, the question posed is whether God's morality is subjective or objective and there is no middle ground between these two alternatives. [2] However, Lewis' concern with the "objective" horn of the dilemma turns out to be unfounded. An objective law need not be external and antecedent to the being that follows it.

To see this, consider the economic law of supply and demand. The truth of this law depends on the actual interactions between people. The law did not precede the existence of people since it depends on what people do. But neither did anyone create the law. Instead it is a discovered generalization of people's behavior. That is, the law of supply and demand describes what people do or, to phrase it differently, people act according to the law of supply and demand. [3] The law is objective rather than subjective because it exists independently of anyone's opinions about it albeit, in this case, not independently of people's behavior.

Similarly the existence of the moral law for God is conditional on God's nature and therefore not antecedent or external to it. Given God's nature, it prescribes what God should and should not do. That is, God is subject to the moral law which he did not create but which nonetheless depends on his existence. Adding the premise that "God is (always) good", the moral law also describes what God does and does not do. As Lewis says elsewhere, "... the Divine Will is the obedient servant to the Divine Reason." [4]

Note: The solution to the dilemma involves other philosophical issues which I haven't explored here but which I take a generally Aristotelian approach to. These include the problem of universals (what does it mean for abstractions, such as the moral law, to exist?), the is-ought problem (how does the moral law derive from a being's nature?) and the argument from morality (does the moral law require God?).

--

[1] "The Poison of Subjectivism" from "Christian Reflections" by C.S.Lewis.

[2] Lewis is aware that he doesn't have a satisfactory solution to the dilemma. He says, "But it is probably just here that our categories betray us. It would be idle, with our merely mortal resources, to attempt a positive correction of our categories - ambulavi in mirabilibus supra me." (Translation: I do exercise myself in great matters, in things too high for me.) However, despite his rejection of the dilemma in this instance, Lewis' general tenor in this essay and other writings is toward the "objective" horn.

[3] The law of supply and command is usually thought of as being true all else being equal. So, for example, when demand increases for a fixed supply of oil, government regulation could prevent the price from rising.

[4] Letter from C.S.Lewis to John Beversluis a few months before his death in 1963. From "C.S.Lewis and the Search for Rational Religion", p295, John Beversluis.

Making good rules

At my kid's school we have an important decision to make when we get to the main gate. Do we go up the stairs to Liam's class first or through the gate to Michelle's class first?

Of course both my children want the opposite thing which is to go to their sibling's class first. So my choice would always end up with one unhappy child and a futile discussion. I sometimes have my one year old Jason with me so that complicates the herding process as well.

So I instituted a very simple rule. Let's go turn about each day. It seemed entirely fair and obvious - what could possibly go wrong?

Many things as it turned out. One problem was that neither of my kids could seem to remember who went first the day before despite my detailed descriptions. Also Joanne sometimes dropped them off, so I would get a conflicting story as to who went first then. And even I would sometimes misremember, escalating their sense of injustice.

So I finally came up with a new rule. If Jason is with us (which happens twice a week) we go to Michelle's class downstairs first. Otherwise it's upstairs to Liam's class. And it has worked perfectly every time. The rule works, I think, because it depends on something immediately observable by everyone and can be determined instantly without requiring discussion. It doesn't depend on memory or a past history of events which is what leads to the competing interpretations and conflicts.

There is a teacher who holds the gate open in the mornings. Our kids know he doesn't know the rule so they like to quiz him on which way he thinks we will go each morning. So instead of the walk into school being a painful exercise, it has now turned into a fun event for everyone.

Free to choose

(This post follows on from Beyond Belief.)

In our everyday lives we are familiar with making choices. We choose what to wear and what to have for breakfast. We are also aware of the constraints on our choices. You can't choose to wear a red shirt if you don't have one.

All familiar and obvious stuff. Until the philosophers weigh in...

Suppose, they say, that all our actions can be fully described in cause-and-effect terms by the laws of physics. Wouldn't that mean that the outcome of our choices is inevitable? And therefore that the freedom of our choices is illusory?

The answer is that, no, in a physically determined world, our choices would be neither inevitable nor illusory.

The hidden assumption in the free-will dilemma is that the determination of an outcome on one mode of description (physics) excludes a determination of that outcome on another mode of description (the intentional, which is expressed in our everyday language of purpose and choice). So for the Determinists free will is impossible. For the Free-Willers physical determinism is impossible. However, as I will demonstrate below, the above assumption is false: the physical description does not compete with the intentional description but instead is logically complementary to it. [1]

As an example of complementary modes of description, consider a computer that calculates the sum of two numbers. At one level of description, the answer that the computer gives is fully determined by the rules of arithmetic. At another level of description, the answer is fully determined by physical cause and effect in the computer. Not surprisingly, since the computer has been programmed to follow the rules of arithmetic, the physical and arithmetic descriptions are complementary, not mutually exclusive.

In a similar vein, our choices are free on the intentional mode of description because there cannot exist, even in principle, a physical description of the future state of our brain that we would be logically compelled to accept as correct. While we would be correct to accept a physical description that includes the change in brain state entailed by our acceptance of it, we would also be correct to reject it. Why? Because our rejection of it would result in a physical brain state that differed from the physical description we were originally considering. Therefore we would also be correct in our rejection of it. So both possibilities are always logically open to us. It's not merely that we wouldn't know whether the physical description is correct or not (an epistemological issue). It's that no physical description of our brain state can possibly exist that would be inevitable for us to choose (an ontological issue). So the shared hidden assumption of the Determinists and Free-Willers is false as a matter of logic. [2]

When we employ our everyday language concepts of purpose and choice, we are operating at a different level of abstraction to that of scientists when they describe the physical laws of the universe. The free-will dilemma is dissolved when we realize that it is our familiar everyday contexts that define our choices and constraints, not the mathematical laws of physics (where intentional concepts are inapplicable). The dilemma only arises when the terms that apply to one mode of description are taken out of their context and applied to a different mode of description.

--

[1] This view is known as Compatibilism. According to a 2009 survey, the ratio of opinions among philosophers is: compatibilism 59.1%; libertarian free will 13.7%; no free will 12.2%; other 14.9%.
[2] This insight is originally from physicist Donald M MacKay ("What Determines My Choice" in his book "The Open Mind and other essays").

The language of illusion

We perceive our surroundings via our sight, hearing and other senses. In most situations, perception is a straightforward process. It's raining. We see that it's raining. We reason that we don't want to get wet so we grab our umbrella.

Mistaken conclusions are discoverable and correctable through the same processes of perception and reasoning. This includes mistakes we associate with the perceptual process itself which we identify as illusions.

In everyday language, we describe illusions by making a distinction between the way things appear and the way things are. When we see a pencil partially submerged in water, we say that the pencil looks bent but is actually straight. This useful distinction enables us to avoid any confusion when talking about illusions.

With illusions there is no "bent pencil" that we perceive in our minds (recall that introspection is not a kind of observation) or perceive in the world. The takeaway lesson is that appearances can be deceiving, not that the "bent pencil" has a ghostly existence of its own that we somehow perceive.

I'll finish with a fascinating example of the color phi phenomenon that manages to combine both the illusion of motion and time travel. In the experiment, a red spot is lit for 150ms and then turned off. A short distance away, and 50ms later, a green spot is lit for 150ms and then turned off. Surprisingly, the observers report seeing a red spot moving to the right, turning abruptly green at the half-way point (before the green spot is lit) and then moving to the final position before disappearing. [1]

--

[1] "Consciousness Explained", p114, Daniel Dennett. The phi phenomenon is the illusion of movement which, for example, we experience when watching a movie consisting of rapidly changing static images.