Mach–Zehnder interferometer Particle or Wave?

This is a continuation of what we have covered in the two-slit experiment. If you have forgotten the main concepts, kindly review them.

We will try to shed some light on what makes up light: particles or waves.

In papers written about this subject, such as of the Mach-Zehnder interferometer, you will often read: "The explanation for this result is that it appears a single photon travels both paths and engages in interference with itself." Language fails us when we are trying to describe a QM system. However, the language of mathematics can compensate.

So first, a few words on the devise in question we will use for this blog. The Mach-Zehnder interferometer is made of a collimated beam which is split by a half-silvered mirror or a beam splitter. The two resulting beams are each reflected by a mirror. The two beams then pass a second beam splitter and enter a number of detectors - in figure 1(a),(b), there are two detectors A and B; in figure 1(c) there three detectors, C being the additional one.

Fig 1

The enigma is in figure 1(a). So before we tackle this situation, we will look at figure 1(c), reproduced below.

Fig 1(C)

To keep track of the different beams, we use E, for beaming going East, S for south, N for north, etc. The incoming beam is labelled I, the outgoing one, O. (See figure 3, below)

After the incoming wave goes through the split beam, (top left-hand corner) we have.

| I > → |E> + |S>

The East going state will reflect at the mirror and then go South, to undergo another split at the bottom right-hand corner, and then go the detector A and B. We can write this whole sequence as,

|E> → |N> → 2^-1/2(a|A> + b|B>)

Here the constants a and b are needed as we must normalize the state vector. This is another reason why we shouldn't take the wavefunction in QM as real. A better language is the state vector using the Dirac notation as we are doing. We normalize it because the state vectors in QM allow us to calculate probabilities, and a simple mathematical rule is that the sum of all probabilities in a given situation adds up to 1. So we need to put another constant for the South going state, which will be reflected by the bottom left-hand mirror, to go Eastward.

|S> → |E> → c|C>

Therefore, | I > = 2^-1/2(a|A> + b|B>) + c|C>

Normalization

< I | I > = a²/2 + b²/2 + c² = 1

By symmetry, we can choose: a = b

a² + c² = 1

Giving us a = b = c = 2^-1/2

Putting this altogether:

| I > = 2^-1|A> + 2^-1|B> + 2^-1/2|C>

A little calculation will give:

Probability at A:

P(A) = │< I | A >│² = 1/4.

Similarly, P(B) = 1/4 and P(C) = 1/2

Particle or Wave

Fig 2

At the heart of the mystery - particle or wave - is what's happening in figure 1(a), reproduced above in figure 2. Keep in mind that the light intensity has been reduced so that only one photon at a time travels through the apparatus. What is seemingly unexplanable are the numbers, which read as: 0% in detector A, and 100% in detector B. Even if the single photon would interfer constructively with itself at the beam-splitter(bottom right-hand corner), after that, each wave should continue along their path, and 50% should go to A, 50% to B. But we don't get that, instead we get 100% at B, none at A. That would mean the wave coming from the southern branch moving from left to right, goes 100% (of 50%)through the beam-splitter(bottom right-hand corner), IOW, doesn't split and continues to B (depicted in fig 1b), while the northern branch, which is now moving downward after reflection with the mirror(top right-hand corner), but now makes an extraordinary 90 degree turn at the beam-splitter (bottom right-hand corner), without splitting to move 100% (of 50%) from left to right to reach B (depicted in fig 1a)!!!

In terms of our state vectors, we can depict the situation in figure 2, with figure 3, below.

Fig 3

We notice that this is very similar to ordinary vectors, depicted below.

Fig 4

We have (we will not normalize because it is redundant as you will see),

| I > → |E> + |S> , Figure 4a

| E > → |S> , at the top right-hand mirror

| S > → |E> , at the bottom left-hand mirror

Therefore,

| I > → |S> + |E>

And at the second splitter,

|S> + |E> → | O > , Figure 4b

Therefore,

| I > = | O >

All the particles will end up at detector B.

By symmetry, if we run the experiment backward:

| O > → |W> + |N> → |N> + |W> → | I >

This is a testimony of the power of the mathematical framework of Quantum Mechanics.

Spooky Action at a Distance and Bell's Theorem Revisited

You have a theorem based on two assumptions:

1. Assumption A (logic)
2. Assumption B (locality)

You design an experiment which violates the theorem. ( Hint: it's a quantum system)

What can you conclude?

Either A is false, or B is false, or both.

But we know that A is false on account that Bell used a mathematical framework for classical system. In classical physics, we need to know the position and momentum in order to know everything about the particle, and these are points in phase space. So set theory, points in set theory to represent position and momentum, and Boolean algebra is the right mathematical framework for classical physics. Call that mathematical framework classical logic. So from that, we can say that Bell's theorem tells us that what we have is a classical system.

But for a quantum system, we need to represent the state of a particle by a vector in Hilbert space, and observables by operators acting on those states, not points from set theory. This is a totally different framework than a classical system. Call that quantum logic. So Bell's theorem applied to a quantum system is not going to work. Violations of Bell's theorem does not prove non-locality, or what was called spooky action at a distance.

Going back to our two assumptions:if A is false, we cannot conclude that B is false. Therefore we don't have any conclusive proof of non-locality.



Thought Experiment


You have to understand that Bell's theorem applies to classical system.

Now, you devise an experiment. You look at your results and they don't fit with that theorem.

You wonder why. You study that theorem carefully and you find that it is based on two assumptions. You don't know which one is false. Is it A, is it B, is it both?

Then some smart physicist called Susskind comes to you, and say, listen, the first assumption is wrong.

You ask why. He demonstrates. The logic applies to classical physics, and then points out that your experiment is about a quantum system.

Would you deduct from this that assumption B is true or false? I think not.

What I'm saying is if you want to investigate whether or not non-locality is a fundamental feature of the universe, you need to forget about Bell's theorem. You are going to need another yardstick from which you can design an experiment that will allow you to investigate that issue.


One More Argument


In Bell's theorem:

We make two assumptions in the proof. These are:

A. Logic is a valid way to reason.
B. Parameters exist whether they are measured or not. For example, when we collected the terms Number(A, not B, not C) + Number(A, B, not C) to get Number(A, not C), we assumed that either not B or B is true for every member.

Consider any measurements A, B and C.

Classical system:

You use Boolean algebra, based on set theory, you get (Bell's inequality):

(1) Number(A, not B) + Number(B, not C) is greater than or equal to Number(A, not C)

Quantum system:

You use vectors in a Hilbert Space, you get (violations of Bell's inequality):

(2) Number(A, not B) + Number(B, not C) is not greater than or equal to Number(A, not C)

You do an experiment, and your results confirm (2), then your only conclusion is that you have a quantum system.

The Universe Was Never a Singularity

At any given time, the radius of the universe was greater than the Schwarzschild radius. Otherwise, the universe would have collapsed into a black hole. The Schwarzschild radius can only be zero if and only if the universe has no mass and no energy. Since the mass/energy of the universe was never zero, its Schwarzschild radius had to be greater than zero, and the radius of the universe, being greater at all times than its Schwarschild radius, could have never been zero. Therefore, the universe was never a singularity.

The Oscillating Universe

Consider that there are two universal principles governing the fate of the universe: one, energy is conserved; two, the total energy is zero. Then as energy is perpetually undergoing a conversion between the potential energy of Dark Energy and kinetic energy of matter, the expansion and contraction of the universe would be a result of that interplay. There would be no Big Bang, no Big Crunch and no Big Rip, and no need for extra dimensions. In a way, Dirac’s sea of negative energy would be vindicated. It also could account why the universe is populated with matter and not anti-matter.

Potential Energy (P.E.), is encoded in Dark Energy, and Kinetic Energy (K.E.) is encoded in ordinary matter ( baryonic + dark). Dark Energy generates a repulsive force, and matter, an attractive force.

The total energy of the universe is zero: − E + E = 0

1st phase: At the beginning of this phase, the potential energy of Dark Energy is at its maximum, while the kinetic energy of matter is at its minimum. As the potential energy of Dark Energy decreases, the kinetic energy of matter increases. What we see is a universe with galaxies moving away from each other. Also, temperature decreases (Cosmic background Radiation). When this phase of the cycle has been exhausted, the potential energy is at its minimum, and kinetic energy is at its maximum.

2nd phase: The kinetic energy of matter decreases ( the universe is decelerating), until its reaches a minimum, after which the universe reverses course, and follows a similar pattern during the 3rd and 4th phases.

Entropy: this is a reversible and adiabatic process

→ dS = 0.

Temperature: In the 1st phase, the universe is hot, its temperature is high. There are two distinct functions for Dark Energy: it acts as a repulsive force, and it acts like a heat reservoir. As its potential energy is converted to the kinetic energy of matter, the universe accelerates. But due to the repulsive force of Dark Energy, it also expands, which has the net effect of cooling it down, (heat flows from matter to Dark Energy).

− E + ∆E → 0 (from maximum negative to minimum)

Note: there are two processes of the energy exchange, ∆E, and we shouldn’t confuse them.

(a) Potential energy is being converted to kinetic energy (acceleration):

For Dark Energy:
a decrease in potential energy,(|P.E.|− ∆E)

For matter:
an increase in kinetic energy, {K.E. + ∆E}

The net result is at any given time:
(|P.E.| − ∆E) + { K.E. + ∆E} = |P.E.| + K.E.

Energy is conserved.

(b) Heat is flowing from matter to Dark Energy (cooling):

For Dark Energy:
an increase in heat, (− E + ∆E)

For matter:
a decrease in heat,{E − ∆E}

But at any given time, the net result is:
(− E + ∆E) + {E − ∆E} = 0

What we see in the 1st phase is that galaxies are accelerating and moving away, and temperatures are going down.

In the 2nd phase, the process occurs in reverse. Initially, the potential energy of Dark Energy has been exhausted. The kinetic energy of matter, after reaching its maximum, is now being converted to the potential energy of Dark Energy.

For the potential energy: 0 + ∆E → |P.E.|
For the kinetic energy: E − ∆E → 0

As the kinetic energy is decreasing, the universe decelerates, and the attraction of gravity dominates, forcing the galaxies to contract.

During that 2nd phase, heat flows from Dark Energy ( it becomes more and more negative) to matter, and the matter in the universe gets hotter and hotter.

What is seen in the universe is that galaxies are moving closer, and temperatures are going up.

In this model, there is no Big Bang, no Big Crunch and no Big Rip, and there’s no need for extra dimensions.

Transition period (T): This is the short period of time when the universe reverses course in direction: the temperature is so high that particles are completely decoupled. However were you an electron, assuming an electron can survive such extreme temperatures, you would see a period in which every thing has been pulled apart and rapidly moving closer to each other( the ending of 4th phase of the old cycle) and soon afterwards, particles starting to rapidly move away, the temperature going down by a very large factor, allowing for recombination (the beginning of 1st phase of the new cycle).

Inflation: This is a result from the thinking that there is a singularity (●) and from that point, one sees a greater expansion than what really happened – which was really a short period of rapid contraction followed soon after by a short period of rapid expansion. This has a net effect of making the universe to appear as if expanding exponentially.

Counter-intuition: You would think that as the energy flows out from matter to Dark Energy (during the 1st phase, for instance), the matter in the universe would slow down. Instead, its kinetic energy is increasing ( potential is converted to kinetic), therefore it is accelerating. Because of the repulsive force, galaxies are moving away, the net effect is that the universe cools off. What we have is that energy, through the interplay between an attractive and a repulsive force, is doing an oscillation through its conversion. Energy is conserved. At the same time, while heat is exchanged,the total energy is zero, and. What can be measured -- the expansion (contraction), the cooling (heating) of the universe -- is a result of these two laws.

Final notes:

(i) Dark Energy had been previously discovered twice. First in Einstein’s General Theory of Relativity, it was called the Cosmological Constant. It was initially used as a factor to stabilize the universe. Unknown to Einstein, the universe would be discovered by Hubble a decade later to be expanding. It was also discovered by Dirac, which is no surprise, as one takes the Einstein energy equation, quantize it and you get the Klein-Gordon equation. Dirac saw in that equation the asymmetry between time and space, and took the square root of it. The pay off was that the spin is encoded into that equation. But also, to prevent energy from sliding into infinite negative energy levels, which would have nixed the idea of a ground state and undermined Quantum Mechanics, Dirac came up with the idea of a sea of negative energy which is fully occupied. And with that, he correctly reasoned that one could borrow from that sea and create anti-matter, a prediction which was confirmed shortly after with the discovery of the positron. But at the times, no one knew what to do with this sea of negative energy. And now, we know better. In this model, the very nature of Dark Energy is potential energy on a global scale. But locally, it can also be seen as anti-matter, but only under the right circumstances. The existence of anti-matter is brief, and this accounts why the universe is populated by matter and not anti-matter.

(ii) Historically, the flaw in a perpetual motion of an oscillating universe resided in the 2nd law of thermodynamics which said that an increasing entropy would eventually bring it to rest. In this model, the total change in entropy is always zero. This is restated as: the total energy of the universe is always zero, which is a stronger statement – the popular version being, “there are no free lunches.” That the change in entropy can be zero had already been discovered in reversible, adiabatic systems. However, since most of the systems we observed are typically irreversible in nature, the second law has been mostly stated with the notion that the entropy always increases, often leaving aside that it can be non-increasing. It turns out that Dark Energy is exactly the means needed by which this process can that place.

iii) Locally, the Higgs field gives matter its mass, providing matter with the central character in the evolution of the universe. Globally, the universe is the stage on which this kinetic energy of matter is seen. While Dark Energy fills the vacuum, it is the field that provides potential energy on a global scale for the universe to oscillate without collapsing into a singularity.

Why our present theories collapse at Planck scale

Suppose we want to probe the Planck scale. To do that we would need a particle of a certain mass m.

According to quantum physics, for every particle, there is a wavelenth associated with it.

(1) λ = ℏ/mc

According to General Relativity, at Schwarzschild radius, when the mass collapses to a black hole:

(2) r = 2mG/c²

Equating (1) and (2), and solving for m, we get,

m_collapse = (ℏc/2G)^½

But the Planck mass is m_pl = (ℏc/G)^½

We can see that m_pl > m_collapse, so the probing particle will have a mass that will precipitate its collapse into a black hole in our attempt to probe at the Planck scale.

The implication is that unless there is a theory that can circumvent QM + GR, we can know absolutely nothing of what is taking place at Planck scale, as if Nature has shut its doors to our inquisitive curiosity.

Entropy, information and gravity

Recall that the overall entropy of the universe is always increasing. You could have an "isolated system", like the earth for instance, in which the entropy could decrease, but when you enlarge that system, earth + sun, the total entropy is increasing. So looking at the earth by itself is really not a good idea in defining what is considered to be an isolated system -- a common mistake people make.

Now, if we take a system in which gravity plays no role, then generally speaking, an increase in disorder corresponds to an increase in entropy. For instance, an ice cube left on the table will melt to water and then evaporate as water vapour.

Ice (high order or low disorder, low temperature) → water vapor ( high disorder, high temperature).

We see from that equation, that going from low temperature to high temperature, entropy increases.

However, when gravity is factored in, the reverse is true. For instance, at the Big Bang, matter was at a very high temperature. You would think that like the water vapour, the universe had high disorder. But it didn't. We will show below that such a system has low entropy (read low disorder) and as it expands, its temperature decreases and its disorder increases.

Universe at Big Bang (high order or low disorder, high temperature) → Universe at present times (high disorder, low temperature).

Here, we go from high temperature to low temperature, and entropy increases.

Black holes -- as well as any star, our sun included -- are like the universe. They are in a state of low disorder. With Hawking radiation -- our sun radiates mainly electromagnetic radiation -- they are moving from low disorder to high disorder, that is, entropy is increasing.

Modern theory looks at entropy as the lack of information. So you can substitute in the above equation, the following:

low disorder (high order) → high information

high disorder (low order) → low information.

As the universe is expanding, our knowledge of the universe is decreasing -- entropy (or lack of information) is increasing, that is, we are becoming more and more ignorant. At first sight that might sound paradoxical, but think of it this way, as the universe grows bigger and bigger, it is more difficult for us to keep track of every particle in the universe. In fact many galaxies are so far way, that the light from them will never reach us. And this situation gets only worse as time goes by.

Gravity

When an object is orbiting around a larger one (say a satellite around the earth, for instance) and changes its orbit, it will do that by emitting away some energy. The result is that it will move to a smaller orbit, but its speed will increase. Hence its kinetic energy has increased, even though it radiated some energy away. The reason being the case is that the gravitational potential energy is negative.

E = K.E. + P.E.

E = ½ mv² – GMm/r

So while the kinetic energy increases, the radius being smaller, the potential has become more negative. In absolute value, the change in potential energy is twice the change in kinetic energy. So whatever gain in kinetic energy, there is twice the loss of potential energy.

ΔE = Δ K.E. + ΔP.E.

ΔE = Δ K.E. – 2 Δ K.E.

= – Δ K.E.

But K.E. = NT, where T is the temperature, and N is any positive constant.

So we must have

ΔT = -ΔE /N

But from the definition of specific heat (c),

ΔE = cmΔT, where m is the mass.

This means that gravitational bound system would have a negative specific heat!!!

To rescue this situation, we must conclude that entropy increases when the system decreases in temperature. The less energy the system has, the higher the temperature. This feature of gravitationally bound systems makes them tricky. Only systems with positive specific heat can be in thermal equilibrium with their environment. So gravitationally bound systems can never be in thermal equilibrium with their environment! They always want to keep shrinking, thus losing energy and, by the 2nd of law of thermodynamics, they increase the entropy. In the satellite-earth system, the satellite will emit energy as its orbit degrades into a smaller one.

NOTE: One might think, is the earth orbit shrinking and will it crash into the sun? The earth's orbit shrinks by 3.5×10⁻¹³ m per year. However, the effect on the size of the Earth's orbit is negligible over the age of the universe. Unfortunately, in the case for the many satellites that humans have launched and are orbiting the earth, as time goes by, most of them will have their orbit decaying in a much shorter time, and will meet their fatal fate by crashing into the earth.

Why FTL violates causality

Suppose we have two observers, one in relative motion with respect to the other. We can represent this in fig 1 with one of the observer at rest -- frame of reference in black -- while the second one in motion -- frame of reference in blue. The line cutting in the middle represents the speed of light, c = 1. In particular, from the point of view of a stationary observer, an observer moving at constant velocity has a coordinate frame whose space and time axes are “tilted” towards the light cone (blue).

Two events a and b are silmultaneous in the moving frame -- they both lie on the x' axis, that is, t' = 0. But for the stationary observer, these are read off at t_a ≠ t_b. Therefore these two events are not simultaneous. Welcome to the world of Einstein's theory of Special Relativity.

But now suppose that we have invented a device that can send signal at speed greater than c, and to make it easy for illustration purpose, we take that speed to be infinite, that is, sending a signal at P arrives instantaneously as Q. This is indicated by the red line in fig 2.

In fig 3, events P and Q are simultaneous in the stationary frame ( X-T frame), while events Q and R are simultaneous in the moving frame ( X' - T' frame). So one could send a signal to himself, which would arrive in his past!

Or to put in another way, I would receive signals from my future self. This would violate causality.