Friday Science: Susskind 4a: Unitarity

Seventh installment summarizing Susskind's, Quantum Mechanics: The Theoretical Minimum.

Chapter 1: Dirac was much smarter than I (introducing linear algebra).
Chapter 2: Quantum States (a.k.a., more linear algebra)

Chapter 3a: Linear Operators
Chapter 3b: Eigenvectors
Chapter 3c: Hermitians and Fundamental Theorem of QM
Chapter 3d: Principles of Quantum Mechanics
Chapter 3e: 3-Vector Operators
Chapter 3f: Spin Polarization Principle

Now that I'm done reviewing Hawking, I thought I would return to wending my way through Susskind's book. Here beginneth chapter 4.

4.1 Classical Reminder
A state is the way something is at a particular point in time. The main rule for how states change in classical mechanics is deterministic. If you know the formula, you know what the next state is going to be. The second rule is reversibility. If you know the state now and the formula for change, then you know what the previous state was too.

If two identical systems have the same state at some point in time, then their past and future is the same as well. We call this "unitarity."

4.2 Unitarity
So consider a closed system. Let's use the Greek letter psi to indicate the quantum state of something: |ψ⟩ . To say that "the state was |ψ⟩ at time t, we will use the notation |ψ(t)⟩ . In a sense, this notation |ψ(t)⟩ represents the entire history of the system.

Assuming a system has unitarity, we can use the operator U to say this too:

|ψ(t)⟩ = U(t)|ψ(0)⟩ 

The entire history of the states of a system is this "time-development operator" producing a series of states that start at time 0.

4.3 Determinism in Quantum Mechanics
The development of a state vector in quantum mechanics is deterministic just like in classical mechanics but with one very significant difference. In classical mechanics, determinism tells us the result of the next experiment with certainty. In quantum systems, it tells us the probabilities of the outcomes of later experiments.

4.4 Closer look at U(t)
1. This time-development operator in quantum mechanics must be linear. That means that for every time you put in, you get one quantum state out and the relationship between the two develops at a constant ratio.

2. The unitarity operator also implies that if two basis vectors are orthogonal (are distinguishable), then they will always be orthogonal. This is called "the conservation of distinctions." This means that, for example:

⟨ψ(t)|Φ(t)⟩ = 0

if these two functions are orthogonal.

3. Susskind then shows that for unitary operators

U^†U = I

where U^† is the Hermitian conjugation of U [1] and I is the "unit matrix." The unit matrix is one where, when multiplied by something, results in the same matrix. It is a matrix with all ones down its diagonal and zeros everywhere else.

It has the equivalent result to the Kronecker delta δ_ij, which yields the value 1 when two things with the same basis vector are multiplied but 0 when orthonormal basis vectors are multiplied.

4. This adds a fifth principle to quantum mechanics. The evolution of state-vectors with time is unitary.

[1] As a reminder, an operator is Hermitian if the matrix version and its transposed version (where you interchange the rows for the columns) yield an equivalent result.

Friday Science 3f: Spin Polarization Principle

Seventh installment summarizing Susskind's, Quantum Mechanics: The Theoretical Minimum.

Chapter 1: Dirac was much smarter than I (introducing linear algebra).
Chapter 2: Quantum States (a.k.a., more linear algebra)
Chapter 3a: Linear Operators
Chapter 3b: Eigenvectors
Chapter 3c: Hermitians and Fundamental Theorem of QM
Chapter 3d: Principles of Quantum Mechanics
Chapter 3e: 3-Vector Operators

Finishing up notes on chapter 3.

3.8 The Spin-Polarization Principle
Any state of a single spin is an eigenvector of some component of the spin.

I wish I more fully understood this principle, but I will do my best. It seems to me it is saying that you're going to get a +1 somewhere.

〈σ_x〉² + 〈σ_y〉² + 〈σ_z〉² = 1

This type of bracket indicates what is called an "expectation value" or the average value of a measurement. The square of the expectation value is the probability of finding a 1 there. So there has to be a 1 somewhere or the probability has to total one.

So, given any state ∣A〉 = α_u∣u〉 + α_d∣d〉

There is some direction 𝜎 ⃗∙𝑛 ̂ ∣A〉 = ∣A〉

3.7 An example
If I had fully followed the matrix analysis of the previous sections, I'm sure this section would be delightful. I get the general sense that he is playing out probabilities in a spherical framework. I generally understand spherical coordinates and the chart on p.89. But I think I'll skip summarizing this section and call chapter 3 concluded.

Friday Science 3e: Three-Vector Operators

Seventh installment summarizing Susskind's, Quantum Mechanics: The Theoretical Minimum.

Chapter 1: Dirac was much smarter than I (introducing linear algebra).
Chapter 2: Quantum States (a.k.a., more linear algebra)
Chapter 3a: Linear Operators
Chapter 3b: Eigenvectors
Chapter 3c: Hermitians and Fundamental Theorem of QM
Chapter 3d: Principles of Quantum Mechanics

Much is still not clicking but let me finish what I can of chapter three.

3.5 A Common Misconception
1. Measurements in quantum mechanics correlate with operators. However, the two are not exactly the same. Measurements come up with definite answers. For example, if you are measuring a particular spin vector, it will either be 1 or -1.

By contrast, the operator has to do more with the probability of a certain outcome. Operators are mathematical rather than actual. They are the tools used to calculate eigenvalues and eigenvectors. We use them on state vectors like "up" "down" "right" "left" "in" and "out." And the result of his operation is another state vector combination that may involve square roots and imaginary numbers--things you will never get in an actual measurement. The actual measurement is always either 1 or -1.

3.6 3-Vector Operators Revisited
2. So Susskind distinguishes three types of vector in this section. The first is a 3-vector space like we use in ordinary directions in life (two miles south, then a mile east, on the sixth floor).

Then he's been talking about state vectors like up, down, right, left, in, out. These are metaphors, I think.

Now he speaks of spin components x, y, and z. He calls these operators, written as matrices. They are the three measurable components of spin. He calls them a new kind of 3-vector, a 3-vector operator. I don't seem to understand, but I'm going with it.

3. Now what if we want to measure spin in any direction sigma n, where n is the direction? Then we can break down the spin in this direction to

σ_n = σ_xn_x + σ_yn_y + σ_zn_z

So if we use the Pauli matrices from the previous post for the components of sigma, we can express the spin in that direction as:

Susskind does some matrix voodoo to combine all these into one big matrix.

Apparently, if we know the eigenvectors and eigenvalues of this particular σ_n, we can use this matrix to calculate all the probabilities for all the outcomes of our measurements of the spin.

Friday Science 3d: Principles of Quantum Mechanics

Sixth installment summarizing Susskind's, Quantum Mechanics: The Theoretical Minimum.

Chapter 1: Dirac was much smarter than I (introducing linear algebra).
Chapter 2: Quantum States (a.k.a., more linear algebra)
Chapter 3a: Linear Operators
Chapter 3b: Eigenvectors
Chapter 3c: Hermitians and Fundamental Theorem of QM

1. Again there is the sense that if I can just make it a little further, he'll connect this stream of math to something concrete so that all the rest will click. I feel like I'm reading 1 John.

Principle 1: Observable quantities in quantum mechanics are represented by linear operators. These have to be Hermitian as well.

Principle 2: The possible results of a measurement are the eigenvalues of the operator that relates to that observable. If a system is in the eigenstate ∣λ〉 , the result of a measurement has to be λ .

Principle 3: Distinguishable states are orthogonal vectors.

Principle 4: The probability of observing a value λ is 〈A∣λ〉² That is the probability of observing a particular eigenvalue is the square of the overlap between the eigenvalue and that state in general.

2. So Susskind uses the spin operator as an example. A spin operator provides information about the spin component in a specific direction. There is a spin operator for each direction in which the measuring apparatus can be oriented.

So he asks what an appropriate "spin operator" might be for the "up-down" aspect of spin. For up, the value will be one for up and zero for down. For down, the value will be zero for up and -1 for down. This corresponds to the following matrix:

z matrix (up down)

This satisfies the conditions: 1) it represents one component of the spin, 2) the possible results are +1 and -1. These are the eigenvalues of this matrix. 3) up and down are orthogonal.

3. He derives the matrices for the "right left" and "in out" components as well. These three matrices constitute the "Pauli matrices."

x matrix (left-right)

y matrix (in out)

Friday Science 3c. Hermitians and Fundamental Theorem of Quantum Mechanics

Fifth installment summarizing Susskind's, Quantum Mechanics: The Theoretical Minimum.

Chapter 1: Dirac was much smarter than I (introducing linear algebra).
Chapter 2: Quantum States (a.k.a., more linear algebra)
Chapter 3a: Linear Operators
Chapter 3b: Eigenvectors

More on chapter 3. I increasingly get the sense that this book should have been written somewhat in the reverse order that he did. Typical linear, building block thinking. Most human minds--especially those this book is allegedly written for, like me--work on a "need to know" basis. That's how this book should be written.

1. Made some progress this week in the book. Think I'm further than I've ever been in it, understanding more than I ever had. Probably could handle a re-read. Since I'm making good progress, I'll try just to jot down some notes.

A "Hermitian" conjugate is like the complex conjugate of a matrix. You do two things to a matrix to find its Hermitian conjugate:

• Interchange the rows and columns (so m₂₃ becomes m₃₂)
• Complex conjugate each matrix element.

A Hermitian conjugate is denoted by a dagger. So the Hermitian conjugage of M is M^† . The matrix might have a T in its upper right hand (for "transposed").

• So you might say that M^† = [M^T]*    (transposed and conjugates).
• So if M∣A〉 = B then 〈A∣M^† = 〈B∣

2. A Hermitian operator is one that is equal to its Hermitian conjugate: M = M^†

The eigenvalues of a Hermitian operator are all real.

3. We now get to what Susskind calls the fundamental theorem of quantum mechanics. It amounts to this: "Observable quantities in quantum mechanics are represented by Hermitian operators" (64). Another way to put it is that "The eigenvectors of a Hermitian operator form an orthonormal basis."

Here is my interpretation of how he unpacks it:

• The possible vectors for a Hermitian operator are all of its eigenvectors and their sums.
• The unequal eigenvalues of a Hermitian are orthogonal.
• Even equal eigenvalues can be analyzed as orthogonal. In other words, two eigenvectors can have the same eigenvalue. This is called "degeneracy."
• If a space is N-dimensional, there will be N orthonormal eigenvectors.

4. The Gram-Schmidt procedure is a procedure for teasing out orthonormal sets that relate to degenerated eigenvectors with the same eigenvalues. Here is the procedure:

• Divide vector one by its own length to get the first orthonormal basis of unit length.
• "Project" the second vector onto that unit vector by taking the inner product with it. 〈V₂∣v₁〉. 
• Subtract this from the second vector.
• Then divide the result by the length of the second vector to get an orthonormal basis for the second vector of unit length.

I don't entirely follow, but I'm making progress.

Friday Science: 3a. Linear Operators

Third installment reviewing Susskind's, Quantum Mechanics: The Theoretical Minimum.

Chapter 1: Dirac was much smarter than I (introducing linear algebra).
Chapter 2: Quantum States (a.k.a., more linear algebra)

1.  So on to chapter 3 (a.k.a., even more linear algebra). This is really as far as I've gotten in the several times I've started plodding through this book. This week I want to summarize a first few pages from chapter 3. It will probably take two more weeks to finish the chapter.

The first half of the chapter is a mathematical interlude. Once again, I think these interludes are most effective after you have introduced a problem you need to solve. Then the math makes sense as a way to solve the problem. Oh well.

2. I think I'm beginning to get a better sense of what bras and kets are. That it is so simple to say is part of my frustration with Susskind's pedagogy. In laypeople's language, a ket like ∣A〉 is a collection of complex numbers (I could explain complex numbers). By convention, they are written in an up and down matrix like this:

We still don't really know why we would want such a collection, but we are calling this a vector.

Bras are written as horizontal collections of complex numbers and are the complex conjugates of the bra equivalent (I could explain complex conjugates). We call them vectors too. They are written like this:

The "inner product" of a bra and a ket is simply the matrix multiplication of the two (I could explain matrix multiplication).

3. Machines and Matrices
So linear operators are basically matrices that bras and kets are multiplied by using matrix multiplication. John Wheeler, a famous twentieth century physicist, called them "machines." Again, Susskind hasn't really given any sense of why we would need these or when we would use them.

But you basically use them to "operate" on bras and kets. For example, here's a linear operator that you might multiply a bra or ket by:

"Operating" this on a bra or ket is like plugging a number into an equation, except we are multiplying a bra or ket by this matrix.

In notation, we might say M∣A〉 = ∣B〉 . The operator M takes the input and spits out the output. ∣A〉 and ∣B〉 are kets.

4. Linear operators 1) relate to observable features in quantum mechanics (=real not imaginary stuff), 2) are like functions--you need to get an output for every imput, 3) multiplying the input by something needs to get the output multiplied by that something, and 4) whether you do the machine on the sum of vectors to begin with or do it on the sum of the outputs, the result should be the same.

Some of the "observables" you use these in relation to include: position of a particle, its energy, its momentum, its angular momentum, or an electric field at a point in space.

I read more than the six pages this post covers (51-56). If I get a chance before next Friday, I may blog some more, but gotta fly.

Friday Science: Susskind's Quantum Mechanics

1. About two years ago, I bought Leonard Susskind's Quantum Mechanics: The Theoretical Minimum. I'll confess that I have found it an incredibly frustrating book. As I've read and reread the first few chapters, I have the repeated feeling that all this book needs is a few more paragraphs in each chapter--and maybe some rearrangement of the order of topics--and it would be incredibly helpful. It seems that he has followed a logical order but not a good pedagogical order.

It's like you're in one part of the forest and he's telling you about a set of trees in another part of the forest. And he's not even talking about trees in his part of the forest that are next to each other, but there's one tree he's seen that relates somehow to another important tree he's seen. But he's none too clear even about how those two trees are connected to each other... in some unspecified part of the forest you're not in.

What you need is directions to get from your part of the forest to the part he's in. Then you need to know how to get from one of the trees he's mentioning to another. This height of unnecessary confusion makes me angry, because it's completely avoidable. I'm convinced I could do much better and probably will.

For the record, I don't think it's intentional. I think Susskind really wants to be clear. He just knows the forest too well to tell someone in words how to get around in it, at least someone who's never taken a course in linear algebra. I've wondered if the jokes at the beginning of the chapters are revealing. They seem to demonstrate an inability to grasp what is funny.

But I want to force myself through the book. Richard Feymann once told his sister to read and reread the math and science she didn't understand. I'm convinced this is the way to go with many difficult subjects and authors. So here I go again with Susskind.

2. Chapter one should not be the first chapter. At least a great deal of what is in here should not be first. Most people need to know why they need to know something for the something to stick and make sense. So I have come to realize that there is some linear algebra in this first chapter. It uses notation that Paul Dirac introduced to quantum mechanics I think in the 1940s.

Wrong place to begin. He's thinking. We learn bras and kets, then we use them in later chapters. But bras and kets make little sense when you have no idea what they're for. I know complex numbers, but for someone who doesn't, it would be better to introduce them when we need to know them. Show us the problem they help solve and introduce them there.

3. So what is helpful to take away from chapter one at the beginning? Here is some stuff from the beginning of the chapter:

• The idea of state is fundamental to quantum physics. For the moment, let's talk about the most fundamental state as being either on or off, +1 or -1. Let's call this "two-state system" a one bit system, a quantum bit or "qubit."
• We could call this "either on or off" the quantum spin. It's not a literal spin.
• Experiments are never gentle. You measure one thing, you mess up everything else. You've lost information from the other place because you've chosen to measure this place. (think Heisenburg's Uncertainty Principle)
• What is predictable on the quantum level is not the individual outcome of some measurement, but the statistical average. Individual outcomes are not predictable, but the averages are.
• The quantum mechanical notation for the statistical average is Dirac's bracket notation: 〈Q〉 .

4. Now he gets into some linear algebra.

• The "space of states," the possible values or states of something is a "vector space" in quantum physics. (linear algebra) Another name for such a "space" is a Hilbert space. There could be an infinite number of elements. This is all very abstract. For the moment, I'm just picturing a box you put stuff in, and different boxes will only take a certain number of things.
• The elements of a vector space are called kets or ket-vectors. The notation Dirac used for these is ∣A〉
• The elements of a ket are often a column of complex numbers. We are being set up for matrix multiplication. 
• The "row" matrix that is pit against the "column" matrix of the ket is the bra. The bra looks like this: 〈B∣ . 
• bra-ket. First in the row multiplied by the first in the column and so forth. The inner product of two of these vectors is the result of this sort of operation.
• A vector is normalized if its product with itself is 1.
• A vector is orthogonal if its product with itself is 0.
• The dimension of a vector space is the maximum number of orthogonal vectors in that space. These vectors are orthonormal bases in relation to each other.
• Finally, there is something called the Kronecker delta. As far as I can see, he never tells us what this is. I know the name from somewhere else. The Kronecker delta is symbolized as δ_ij . This function is 0 if i and j have different values and 1 if i and j have the same value.

Why do we need to know these things? He doesn't tell us. Very frustrating.

Friday Science: Particles Separated at Birth

Another chapter down in Brian Greene's, The Fabric of the Cosmos.

My first two summaries were:

a. Overview
b. Spinning Space Buckets
c. Relativity and the Absolute

1. This chapter seemed a lot longer than it needed to be to me. Hopefully I can give the gist fairly quickly. When you run a water wave through two openings, you will get an "interference pattern" on the other side of the openings. The same happens with light. When you shine laser light through two slits, you get the same interference pattern.

What is very, very strange is that if you take an electron beam and shoot electrons one by one, slowly at those same two slits. If you shoot them, each one separately so they do not interact with each other, over time the very same interference pattern will emerge. So electrons, photons, all matter may be made up of particles, but those particles behave like waves.

The kind of wave it is, Greene helpfully points out, is a probabilistic wave. That is to say, it is because particles have a greater or lesser probability of being at a particular place when interacted with, over time their interaction with the slits plays out as a distribution of lines that fits those probabilities.

2. Werner Heisenberg showed in the late 20s that you cannot measure both the position and velocity of a particle accurately at the same time. If you measure the position with precision you can't measure the velocity and vice versa. This also applies to a number of other atomic features, such as spin.

Einstein engaged in a longstanding debate with the quantum mafia led by Bohr. Einstein couldn't bring himself to believe what has more or less turned out to be true. Particles don't actually have a definite position or velocity until you measure them. The nature of the quantum world is probabilistic. There is a greater or lesser probability that an electron is somewhere. It's not that it is somewhere and we just don't know exactly where. It's that it isn't exactly somewhere.

3. Einstein and a couple colleagues unintentionally advanced this discussion with a thought experiment that was later carried out. He suggested that if two twin particles parted with a correlated identity, as is often the case, then by measuring the position or the velocity of the one you could indirectly infer the position or velocity of the other.

This seems like common sense. What you do to the one doesn't affect what you do to the other, so you can measure the one and not disturb the other. David Bohm extended the Einstein thought experiment to the spin of a particle. In theory, if you measure the spin of a pair of correlated particles, you should be implicitly identifying the spin of the other. [1] In other words, the other one would have a definite position and velocity even without measuring it, contrary to what the Copenhagen mafia insisted.

4. In the 1960s, Jon Bell came up with a way to see if Einstein was correct and in the 70s and 80s, it became possible to test it. He determined that if you randomly measured the spin of two correlated particles in relation to more than two possible states, you could determine whether both particles had a definite spin to begin with. If you randomly measured the spin at three different angles for both particles, those measurements would agree more than 50% of the time if both of them had a definite spin to begin with.

Some of the best tests took place in the early 80s by the French scientist Alain Aspect. He showed that the detectors did not show that the spins agreed more than 50% of the time. What they showed was rather astounding.

• If Einstein had been correct, they would have agreed more than 50% of the time. The implication would be that the particles had a definite state before measurement, as Einstein thought must surely be the case.
• If the quantum mafia had been completely right, the measurements would have agreed less than 50% of the time. [2] The implication would be that the particles had an indefinite state before measurement and randomly took a state when measured.
• What happened is that they agreed exactly 50% of the time. The implication was that they had an indefinite state before measurement but both particles took on the same state when one of them was measured.

The unexpected result, which is one of the most striking findings in all of the history of science is that what you do to a particle in one place, if that particle correlates to a particle somewhere else, you do to both particles. Many aspects of particles are actually indefinite in the first place, but if you interact with one and make it definite in some respect, you make any companion particle definite as well, no matter where it is in the universe.

5. This is called quantum entanglement. What you do to a particle here can affect a particle there, no matter where "there" is. In a sense, there is no such thing as "locality" in space. There is no "here" that is distinct from "there."

It's not that one particle sends a message somehow to the other, correlated particle. They rather have a unity that transcends locality. Special relativity is not violated. Nothing moves faster than the speed of light. It's just that there is a synchrony that transcends space.

[1] BTW, Bohm fled the US in the middle of the McCarthy nonsense and ended up in England at the end of his life. I hope America will never have a witch hunt like that again. Just think of how many brilliant minds Hitler lost in the middle of his ideological nonsense. No country can afford to lose its scientists for whatever stupid reason the public or politicians come up with.

[2] The Copenhagen circle with people like Niels Bohr, Werner Heisenberg, and Wolfgang Pauli were positivists in philosophy. They didn't consider anything to be real if you couldn't measure it. Their way of explaining the uncertainty principle is deeply unsatisfying to me. Although Einstein proved to be wrong, his objection to them was perfectly valid. Just because you can't measure something doesn't necessarily prove it doesn't exist.

Friday Science: Fabric of Cosmos 1

1. I had been reading a book called, Our Mathematical Universe. I got through the part of the book that is generally accepted by physicists of the universe. But I came to realize that most view the rest of the book not only speculative, but perhaps bordering on the irresponsibly speculative.

So I've switched to another book on the current state of physics: Brian Greene's The Fabric of the Cosmos. Now he is also speculative. He obviously likes string theory and the idea of multiple universes. I'm not real fond of either but at least these are well-trodden paths.

So I thought I'd dawdle through this book for a few Fridays. Chapter one is called "Roads to Reality: Space, Time, and Why Things Are as They Are."

2. The progression of the chapter is roughly:

• Classical Reality
• Relativistic Reality
• Quantum Reality
• Cosmological Reality
• Past and Future Reality

The first half of this material will be familiar to the science enthusiast. He starts with Newton's sense that space and time are fixed entities in which we move. Einstein transformed our understanding here, for space and time become adjustable.

Entering the quantum reality apparently requires us to throw all our intuitions out the window. Here we encounter an idea I believe I first saw in Richard Feynman. Human intuitions were formed to help us survive and thrive in the macro-world. (I think you might say so whether you are speaking of how God made us or of how evolution developed us). The implication is that our "common sense" and our intuitions have no point of reference for the quantum world.

So the math seems to work, but no one really knows what it means. There are aspects of math itself that are are like this. Take Euler's famous equation from the 1700s: e^iπ - 1 = 0. What does it mean to raise something to the power of the square root of negative 1? I don't have a clue, but it works.

In the quantum world, at least so far, you cannot predict things. Rather, each event has a probability of happening. The universe is not determined. It is a game of chance.

3. When Greene gets to his section on cosmological reality, he covers some of the bases that I had been reading in Tegmark. One cosmological reality is the fact that the arrow of time only points in one direction. In theory, it would not have to be so. But the current sense of things is that something that happened very early in the history of the universe flipped the switch that makes time unidirectional.

Then he covers the big bang, the idea that the universe expanded rapidly into something like its current form from a much smaller version. He also mentions ideas new to me from Tegmark, which apparently have been around since the 70s and 80s--inflationary cosmology. This is a supposed period before the big bang when space itself expanded a million trillion trillion times in less than a millionth of a trillionth of a trillionth of a second.

4. What's missing is a grand unified theory that can reconcile both quantum mechanics and relativity. He seems to like Superstring theory and M theory. I sense increasing disgruntlement with these theories because there is no experimental data to suggest them whatsoever. They are completely hypothetical. Even Sheldon has given up on string theory. :-)

Feynman 7: New Laws of Nature?

And so we reach the end of Richard Feynman's famous lectures, The Character of Physical Law, a series of lectures he gave at Cornell in 1964. The six previous were:

• Gravitation as an example of physical law
• The relation of physics to math
• Great conservation laws
• Symmetry in physical law
• Telling the past from the future
• The quantum mechanical view of nature

1. In the six previous lectures, Feynman had given some descriptions of the principles of nature. But what is nature. What is the something that these are principles of? What is the energy that is conserved? What is the something that has these mechanical laws?

First, that something is matter. Feynman embarks then to give a taste of the panoply of particles that had been discovered by 1964. The "standard model" was then in process of development. In fact, I have another book of Feynman's from the 80s I'd like to blog through sometime called QED, in which he presents some of this material from a vantage point twenty years later.

"All ordinary phenomena can be explained by the actions and the motions of particles" (151). And these particles are present everywhere in the universe. The make-up of matter in far away galaxies is exactly the same as the make-up of matter here.

The many particles that have been discovered--electrons, protons, neutrons, photons, neutrinos, mesons, anti-particles, etc--can be grouped into families. Feynman describes the situation in his time as similar to that of Mendeleev when he was putting together the periodic table and scientists were locating elements on it.

He also mentions a problem that I believe continues even today, which I consider to be Kuhnian "naughty data" just calling for an Einstein or Dirac to solve. Why do so many of the equations of quantum mechanics go to infinity unless you trick them?

2. So how do you find "new laws" of nature? Feynman presents the scientific method. He is worth quoting: "If it disagrees with experiment, it is wrong. In that simple statement is the key to science. It does not make any difference how beautiful your guess is. It does not make any difference how smart you are, who made the guess, or what his name is--if it disagrees with experiment, it is wrong. That is all there is to it" (156).

One interesting feature about the development of twentieth century physics is the fact that studies were sometimes wrong and not immediately recognized to be so. That is encouraging. These individuals who developed relativity and quantum mechanics sometimes presented papers with errors. They weren't like the proofs of geometry. The greatest minds in physics often did not immediately see what was wrong with their experiments or lines of thought. They were, in the end, mere mortals.

Feynman gives a nearly straight line from Karl Popper--"There is always the possibility of proving any definite theory wrong; but notice that we can never prove it right" (157).

3. It is fun to see how Feynman suggests the process begins. First you look in an area where there are problems and unknowns. Then there is a "feeling" around, an intuitive step because you do not know exactly where to look or even perhaps what you are looking for. Let's just say this is not how a lot of people imagine science working.

As the chapter progressed, Feynman makes it clear that he gets regular mail from ignoramic crack pots like me making stupid suggestions in an area about which they are incompetent. Here are some fun quotes from the last part of this chapter:

• "Such remarks are obvious and are perfectly clear to anybody who is working on this problem. It does not do any good to point this out. The problem is not only what might be wrong but what, precisely, might be substituted in place of it" (161).
• "So please do not send me any letters truly to tell me how the thing is going to work. I read them--I always read them to make sure that I have not already thought of what is suggested--but it takes too long to answer them, because they are usually in the class of 'try 10:20:30'" (161-62).
• "The inexperienced, and crackpots, and people like that, make guesses that are simple, but you can immediately see that they are wrong, so that does not count" (171).

He talks about how people tell him to start from first principles. The problem here is that "all the principles that are known are inconsistent with each other" (160-61). It's easy to point out the problems (as I have)--inconsistencies, infinities--but what are you going to substitute in its place? THAT is what is important.

And there are an infinite number of possibilities of these simple types.

4. In the end, Feynman argues, the great discoverers are great guessers. He mentions Newton and Maxwell, the two first greats. Newton's laws were fairly close to the surface. Not so today. Maxwell guessed at the right answers on the basis of a wrong idea. Einstein was driven to resolve paradoxes among existing laws. Quantum mechanics was developed from two completely different starting points.

Here is another key insight of Feynman: "We must keep all the theories in our heads, and every theoretical physicist who is any good knows six or seven different theoretical representations for exactly the same physics. He knows that they are all equivalent, and that nobody is ever going to be able to decide which one is right at that level, but he keeps them in his head, hoping that they will give him different ideas for guessing" (168).

In the end, he does not believe that history repeats itself in physics. The next big discovery will not come in the way it came for Newton or Maxwell or Einstein or Schrödinger. How do you know when it has worked? "Science is only useful if it tells you about some experiment that has not been done" (164). "You can have as much junk in the guess as you like, provided that the consequences can be compared with experiment." "It is not unscientific to make a guess" (165).

When two principles work in a certain area but are inconsistent with each other, how do you find harmony, if you should? "To guess what to keep and what to throw away takes considerable skill. Actually, it is probably merely a matter of luck, but it looks as if it takes considerable skill" (166).

Feynman gives some of his guesses. Here's an interesting one: "I rather suspect that the simple rules of geometry, extended down into infinitely small space, are wrong." In other words, his hunch is that space is not continuous.

5. You can tell that Feynman isn't particularly impressed with a lot of philosophers. However, "the philosophers who are always on the outside making stupid remarks will be able to close in" eventually, as the science gets closer and closer to completeness. Eventually, the unknown of physics will become known, and then the physicists will not be able to "push them away" (173).

The chapter ends with these striking thoughts on the future of physics. "I think it has to end in one way or another" (172). "We are very lucky to live in an age in which we are still making discoveries. It is like the discovery of America--you only discover it once. The age in which we live is the age in which we are discovering the fundamental laws of nature, an that day will never come again."

"There will be a degeneration of ideas, just like the degeneration that great explorers fell is occurring when tourists begin moving in on a territory." "In this age people are experiencing a delight, a tremendous delight that you get when you guess how nature will work in a new situation never seen before." (173).

Why does it work like this? Feynman's feeling is that it is because "nature has a simplicity and therefore a great beauty" (173). Feynman, who of course was a genius the likes of which I have never met, could recognize when he had a breakthrough. "You can recognize truth by its beauty and simplicity. It is always easy when you have made a guess, and done two or three little calculations to make sure that it is not obviously wrong, to know that it is right" (171).

6. The quote at the beginning of the last paragraph is the end of the chapter and the book. Of course, again, these are arguments that have next been extended to God as the great artist and creator, the great designer.

But in my following the flow I have missed what I consider an important point. Many experimental physicists by their very personality as pragmatists may scoff at competing philosophies about what is happening. My post over the previous chapter dipped into some of those debates, which many consider irrelevant.

But Feynman makes it clear that these philosophies can actually be important. These philosophies are "really tricky ways to compute consequences quickly" (169). "A philosophy, which is sometimes called an understanding of the law, is simply a way that a person holds the laws in his mind in order to guess quickly at consequences."

But it may be that one of these approaches to the data gives a slightly better way forward with the unknown. Newton's laws of gravitation worked oh so well except for this tiny discrepancy with Mercury. That gave Einstein a window to completely re-conceptualize the matter. And so, if it were to turn out that pilot waves provided a way forward that indeterminacy does not, even though the results are entirely the same otherwise, that would make it a better theory.

Feynman 6: Quantum Mechanical View of Nature

This is the second to last chapter in Richard Feynman's, The Character of Physical Law, a series of lectures he gave at Cornell in 1964. The five previous were:

• Gravitation as an example of physical law
• The relation of physics to math
• Great conservation laws
• Symmetry in physical law
• Telling the past from the future

1. In this lecture Feynman gives the Bohr interpretation of quantum mechanics, the interpretation of quantum phenomena held by the majority of quantum physicists. Mind you, the majority of quantum physicists are simply followers of a Kuhnian paradigm they learned in school. Much smarter than me, but not the likes of Feynman or Hawking or Kip Thorne. And Hawking is wrong as often as not these days (e.g., on the existence of the Higg's boson).

I suspect most quantum physicists get annoyed at the question of whether the dogma of uncertainty is right. But it smells like classic Kuhn to me. Logical positivism died in philosophy some sixty years ago, but it is still the name of the game among classical physicists. Feynman ends this chapter with a nice touch. It's okay to have biases as long as you're willing to change them given experimental evidence. I'm willing.

But the current situation in physics has Kuhn written all over it. Bohr, it seems to me, was an ideological bully with charisma. When de Broglie proposed that nuclear particles had pilot waves, he was shut down by the Bohr mafia, the clique that ruled the physics roost at that time. John von Neumann claimed in 1932 to have shown that there couldn't be any other hidden variables like de Broglie's pilot waves that go with particles. It's what the Kuhnian dominant group wanted to hear. Case closed.

Except it wasn't. A physicist by the name of Greta Hermann found an error in von Neumann's argument in 1935, a fact ignored till the 1980s. No one wanted to hear.

Similarly, David Bohm in the 1950s was able to solve the problems with de Broglie's original version of pilot wave theory. John Stewart Bell revived Bohm's approach in the 1980s and also clarified why von Neumann's objection didn't work. And now, John W. M. Bush at MIT has shown that analogous phenomenon in fluid mechanics demonstrate the same results as the standard quantum approach. They require more complex explanations, but the results are the same.

2. I checked some of the online physicist response to Bush and it sounds very much like what Kuhn described as the expected reaction of "normal science." "Who cares." "It's just a different interpretation." It yields the same results but the Copenhagen interpretation is simpler. "It's just about what philosophy you feel most comfortable with."

Here I suppose my theology should bias me toward the indeterminant Copenhagen, but my distaste for logical positivism is even greater. Logical positivism basically says that a falling tree doesn't make a noise in the forest unless someone is there to hear it. If you can't observe it, it doesn't exist.

On the other hand, physics has been stuck for a long time. Hawking can talk about a theory of everything but he's got nothin. There hasn't been any real progress made on a unified theory in a half a century. Quantum mechanics and relativity are just as irreconcilable as they were in the 1930s. String theory has produced NOTHING, and Sheldon was smart to give it up (Big Bang Theory).

This situation suggests to me that something needs backed up to first principles, and the Copenhagen bullies seem as good a place to start as any.

3. Of course none of this is what Feynman presents in this chapter. For all Feynman knew in 1964, von Neumann's critique of de Broglie stood. In his words, "That theory cannot be true" (146).

The double slit experiment basically shows a number of seemingly contradictory things (watch the video):

• that electrons go through one of the two slits one at a time (and thus behave like particles)
• that electrons going through two slits produce an interference pattern (an thus behave like waves)

Even if you send the electrons one by one, particle by particle, they will end up producing an interference pattern like a wave. This is a remarkable thing. It's like the electrons know where they need to go to make the interference pattern even though you shoot them one by one.

But if you try to observe which slit each electron goes through, it stops yielding an interference pattern. You can't tell which hole the electron goes through without in effect changing the situation (to detect is to force a different outcome).

4. Feynman again emphasizes that there is nothing that can be understood about this situation. We simply have to accept it. The equations work even though they have no meaning.

"We invent an 'a', which we call a probability amplitude, because we do not know what it means... To get the total probability amplitude to arrive you add the two together and square it" (137).

• Nobody can give you a deeper explanation for this situation. They can only describe it in more detail. So "you can mention that they are complex numbers instead of real numbers" (145). "But the deep mystery is that no one can go any deeper today." 
• Nature herself does not know which slit the electron will go through.

In effect, "the future is unpredictable" (147). "It is impossible to predict in any way, from any information ahead of time, through which hole the thing will go, or which hole it will be seen behind" (146).

4. I am open to the Copenhagen interpretation, mind you. At the beginning of the lecture Feynman warned that intuition and common sense are completely useless in quantum physics because there simply aren't ordinary human world analogies. "I think I can safely say," Feynman said, "that nobody understands quantum mechanics" (129).

Feynman 4: Womanizing Genius

On to chapter 8 of Quantum Man, a biography of Richard Feynman. I only was able to slip in one chapter this week.

So far:
Chapters 1-2: High school, MIT, and Princeton
Chapters 3-5: The Path to a Doctorate
Chapters 6-7: Theorizing the Bomb

One of the most disturbing features of Richard Feynman's life was the way he began to use women after his wife died. He became a notorious womanizer. You hear of professors in the late twentieth century who used the charm of their genius to entice grad students. Thankfully, those days are mostly over, I hope. The increased attention to sexual harassment and sexual ethics in the workplace is something to be thankful for. Christians who scoff at this sort of thing with the label, "political correctness," have no idea how unchristian they are being.

There are some more innocuous stories of Feynman's increasing disregard for the rules of society. He seems to have become a real Cynic, in the ancient sense. One I found particularly entertaining is how he would sometimes sneak into Los Alamos--the high security place where they were building the first atomic bomb. Then he would leave by the front gate with no record of him ever coming in.

He was flamboyant. He loved the Feynman legends that arose about him. He was a showman. Another stunt was when he intentionally convinced several military psychologists that he was mentally unfit.

The physics of this chapter deals with the some eighteen years between Dirac's breakthrough in which he formulated a relativistic version of Schrodinger's wave equation and a conference on Shelter Island off of Long Island, New York in June of 1947. I. I. Rabi described these years as "the most sterile of the century" (119).

The antiparticle version of the electron--the positron, effectively an electron with a positive charge--was predicted by Dirac's equation and found in 1932. But it gave rise to an even greater pool of infinite occurrences, for there was the possibility in QED, Quantum Electrodynamics, that a photon would momentarily split into an electron-positron pair, only to return to a photon. These sorts of possibilities, occurring seemingly randomly, were part of the new quantum reality.

Most of the calculations of the atom in this period seemed to give nonsensical answers that went to infinity. What the theoreticians seemed to need at this time was some hint from experimental data. Krauss notes that Willis Lamb stepped into this void, "one of the last of a breed of physicists who were equally adept in the laboratory and performing calculations" (119).

In 1946 Lamb found a way to measure the spectrum of the hydrogen atom more finely than had ever been done before. These results were concrete, not some theoretical infinite. He presented them on Shelter Island at a conference called, "Conference on the Foundations of Quantum Theory," a conference Feynman would call the most important one he ever attended. Wheeler, Oppenheimer, Bethe, another young physicist superstar named Julian Schwinger--they were all there.

Lamb presented his results. Bethe was so excited he fixed some of the existing equations on the train on his way to his mother's in upstate New york after the conference. He called Feynman immediately. The race was now on to move forward with quantum theory.

Feynman's look at total paths of particles would play a key role. Relativistic problem enter in when you get to talking about the specific time of different particles, since different objects in motion potentially have different time frames. By looking at the overall energy sums and paths, Feynman had a potential way around the problem.

Let me close with a theological aside. One of the reasons there are so many different interpretations of the Bible and so many different theologies is that there isn't always "experimental data" to ground it. Much of individual theologizing and interpretation is, extensively, unbridled speculation. Indirectly, of course, we as interpreters and theorizers of religion are grounded by our concrete circumstances and the cultures in which we are embedded.

Most of us don't realize that these are as powerful drivers of our interpretations and thoughts as the Bible or God--I would say far more influential, actually. We like to think we are speaking for God or proclaiming the Word of God, but much of it is just self-therapy, giving expression to our inner desires and conflicts.

This is why I long to know the original meaning of the Bible, the real meaning it had in its original times and places. History is cold and uncaring. It is a more or less scientific inquiry. It is the most likely meaning we can suggest given the known literary and historical context.

It is not always certain. In fact it is far less certain than many of us Bible scholars like to think. But, at the same time, it at least eliminates quickly a great mass of things thought and said about the Bible within Christendom. It is a kind of experimental grounding to interpretation and thus gives a tangible grounding to theology.

Feynman 3: The Bomb and then Depression

This is my third installment reviewing a biography about Richard Feynman called Quantum Man, one of the greatest physicists of all time.

So far:
Chapters 1-2: High school, MIT, and Princeton
Chapters 3-5: The Path to a Doctorate

Now chapters 6-7.

Chapter 6: Loss of Innocence
Finishing his PhD degree at Princeton was a condition for Richard and his fiancee Arline to get married. And so, even though she was deathly ill with tuberculous--and generating significant friction between Feynman and his mother--they got married. She would die two months before the bomb dropped on Hiroshima.

If you remember from last week, Feynman had helped work on the question of separating Uranium 235 from 238. He had worked with Robert Wilson's team on this, and it had been accomplished. Then next step was to build a nuclear reactor to do the separation, and this was taking place in Chicago with Enrico Fermi and John Archibald Wheeler, Feynman's doctoral mentor.

So Feynman went to Chicago. Soon after he arrived, he blew away the "theory group" by performing a calculation that had eluded them for months (78). Robert Oppenheimer picked Los Alamos as the place where the Manhattan Project would play out, and he picked Feynman to come with the first wave of scientists in 1943.

Oppenheimer was an unusual scientist because, as Feynman himself put it, he "was extremely human" (79). He not only understood the science. He had organizational skills and cared about people. He would notoriously regret the role he played in the creation of the atomic bomb. His words at the first successful testing were from the Bhagavad Gita: "Now I am become Death, the destroyer of worlds." Feynman merely grinned as he contemplated the physical causes of the mushroom cloud and sonic boom.

Feynman, once again working best in conversation, ended up almost by accident in a conversation in Los Alamos with a seasoned theoretician named Hans Bethe. Bethe would bounce ideas off of Feynman, who being very loud could be heard to cry out, "No, no! That's crazy." From Feynman's recollection, Bethe always proved to be right. From Bethe's recollection, Feynman was probably the most ingenious person in the whole division.

Bethe was the scientist who discovered that fusion fueled the sun. Bethe said of Feynman that "he could do anything, anything at all" (87). He was put in charge of the computing division. You have to wonder whether the Manhattan Project would have ended before the war if Feynman hadn't have been there.

For example, Feynman developed a mathematical method for integrating third-order differential equations that was more accurate than what they were doing with second-order differential equations. When several boxes full of the parts of an IBM computer arrived, Feynman and another person managed to put them together before the professionals from IBM arrived. It had never been done before.

Oppenheimer would say of Feynman that "He is by all odds the most brilliant young physicist here" (92).

Chapter 7: Paths to Greatness
As an academic Dean, I think I am somewhat unusual. I absolutely love knowledge for its own sake. My boss often quotes me as saying, "No one loves the irrelevant more than I do." But I am a pragmatist and a realist. In most cases, it doesn't matter how brilliant a teacher is if he or she can't teach. And in most cases, it doesn't matter how excellent your program is if no one is buying.

Of course there are top flight research institutions that are so heavily endowed that their professors can push the bounds of knowledge without a care and survive off of some small number of purely genius students. Feel free to hire me to teach there. But that is just not where the majority of academic institutions are. And, for all your pretense to greatness, most purists seem to have a penchant for self-destruction (and an overestimation of how great they are).

So I won't tell you what I wrote in the margins of this biography on reading about how the chair of Berkeley delayed making an offer to Feynman on Oppenheimer's recommendation. When he finally did, he had the gall to tell Feynman that no one had ever refused an offer from Berkeley's graduate department of physics.

Feynman did. He went to Cornell, who had the smarts to hire him two years earlier and give him a leave of absence while he was at Los Alamos. Stupid Berkeley.

But Feynman himself was pessimistic and depressed. What future was there, now that there was such a bomb. "What one fool can do, another can," he said (93). Indeed, it is amazing seventy years later that the bomb has not been used again.

It was natural that Feynman would feel like he had wasted three prime years of his intellect--the greatest discoveries of a physicist are usually made in one's twenties. His father had died of a stroke a year after his wife Arline. Teaching takes a whole lot more work than most imagine and back then there was no training in how to teach.

Feynman was being showered with praise from all corners, but he felt like his best years were behind him. Others thought he was incredible. He felt stupid. "They were absolutely crazy" (96), he thought of offers from Princeton and the Institute for Advanced Study.

Bethe, upon hearing of Feynman's depression, remarked that, "Feynman depressed is just a little more cheerful than any other person... exhuberant" (97).

There is a great story about how hard it was for Feynman to finally write up his dissertation for publication. Apparently, two of his friends forced him to write it up while he was visiting them in the summer of 1947. They practically locked him up in a room. It was easy for him to express his ideas in conversational form. But to write them down in a beginning to end argument in detail, with everything exactly write. That he found a hard time doing.

[I know a couple people I'd frankly like to lock in a room to crank some of their gems out. On the other hand, one might argue that I would have written more scholarly pieces these last ten years if I hadn't started blogging.]

However, writing it up seemed to get him over a hump. Quantum mechanics began to be more visual for him. For the first time, he began to describe quantum mechanics in the language of sums over paths, with each path having an amplitude. It was a fundamental reformulation of quantum mechanics on which all the quantum mechanics since is based. His next task was to relativize it, to incorporate Einstein's relativity into his new version of the older model.

The rest of the chapter mostly flashes back to Dirac's relativizing of the original quantum mechanical equation of Schrodinger. We hear about the spin of particles called fermions, after Enrico Fermi. We hear about boson particles that don't spin. We hear of Wolfgang Pauli's exclusion principle and Dirac's theoretical discovery of antiparticles.

Meanwhile, Feynman was trying to find a way to picture the overall paths of these particles in a way that incorporated relativity the way Dirac had for a single particle at particular time and momentum...

Feynman 2: The Path to a Doctorate

A couple weeks ago, I started reviewing a biography about Richard Feynman called Quantum Man, one of the smartest physics geniuses of all time. It is incredibly humbling to read about him, a genius among geniuses. Sometimes we college professors slip into thinking we have something upstairs--we need a Feynman occasionally to knock the wind out of our presumptuous sails.

Chapters 1-2: High school, MIT, and Princeton

Chapter 3: A New Way of Thinking
We left off with Feynman at Princeton with John Archibald Wheeler, the one who coined the phrase "black hole." One of the things that struck me in the chapters today is the extent to which Feynman thrived in his early years by having someone to spar with intellectually. There's a certain kind of synergy that great designers or artists or thinkers can experience when they're bouncing ideas back and forth with others who are on the same wavelength. The whole becomes even greater than the sum of its parts. Wheeler was one such partner for Feynman.

Wheeler and Feynman had some crazy ideas together that did not end up in their first form, but turned out to be headed in promising directions. I mentioned in the last post the idea that certain photons might move back in time (he would later argue this of antiparticles). Another idea was that all elements were made up of electrons (quarks came closer later).

In 1941, Feynman the gifted grad student presented to the Princeton physics department professors (rather than to fellow students), with Albert Einstein, Wolfgang Pauli, John von Neumann, and other physics titans present.

We also meet the love of Feynman's life in this chapter, Arline. She would suffer and eventually die of tuberculosis, just as Feynman was reaching the end of the Manhattan Project.

While the theory Wheeler and Feynman cooked up about particles moving backward in time would not pan out eventually, it led Feynman to invent a new chapter in calculus. In the theory they were cooking up, "the path of a particle at a given time is affected by the path of another at a different time" (48). It was here that Feynman turned to Lagrange's principle of least action, mentioned in the previous post: "an object will take that path where the total sum of the difference between its kinetic energy at each point and its potential energy at each point is lowest."

So rather than think of events at specific times that cause events at a subsequent time, Feynman would focus on an overall space-time path.

Chapter 4: Alice in Quantumland
In this chapter, biographer Krauss spends a little time over-viewing some of the basics of quantum mechanics as they had developed in the two decades previous to Feynman's time at Princeton. He starts with Erwin Schrödinger's wave equation. His equation opened the door to the probability of finding a particle at any given place in space at a specific time. Strangely, it is not the equation itself but the square of the equation that gives this probability.

The equation by itself sometimes involves the square root of negative 1, which isn't a real number (literally ;-). It's called an "imaginary" number. But √-1 is all over quantum mechanics.

Because quantum mechanics works with probabilities, there is a sense in which a particle can take all the different possible paths from a to b. Feynman's idea was to look at the paths associated with the probabilities in Schrödinger's equation rather than the probabilities of where a particle is at any moment.

Chapter 5: Endings and Beginnings
At a beer party at a tavern in Princeton, Feynman ran into a European physicist named Herbert Jehle who happened to be in town. They got to talking and Feynman mentioned that he was trying to explore the path of particles using Lagrange's least action principle.  Jehle mentioned a paper that Paul Dirac had written on the subject in 1932.

According to Jehle, next day in the library, on the spot, working the math more quickly than he could follow, Feynman took Dirac's paper to the next level. Feynman would meet Dirac in 1946 and mention it to him. The almost entirely uncommunicative Dirac was said to say, "Oh, that's interesting." A story about another conversation between the two is that Dirac asked Feynman, "I have an equation, do you?"

I suspect that this was a conversation between the two greatest minds in physics in the twentieth century.

It was the moment that birthed what would become Feynman's thesis and his greatest contribution to quantum physics. He would develop a system of drawing probable paths of particles. He would show that his new approach to paths both reduced to Schrödinger's equation over short paths and to Newtonian ones on a large scale.

Meanwhile, World War 2 had started. In 1942, Feynman was tapped to work on the Manhattan Project. Robert Wilson, an instructor at Princeton in experimental physics, showed up at Feynman's office one day and asked if he was interested in helping him develop a method for separating Uranium 235 from Uranium 238. These are two isotopes of uranium--same type of atom but with different numbers of neutrons. U238 is stable (although it can become unstable plutonium). U235 is prone to deteroriate into Barium and Krypton, while releasing free neutrons that can then impact the nuclei of other U235 nuclei, setting off a chain reaction.

At first Feynman wasn't interested, but in the end couldn't turn down the opportunity. Wheeler thought he was close enough to finish his doctoral thesis in 1942 and urged him to turn it in. Feynman "had re-derived quantum mechanics in terms of an action principle involving a sum" (an integral). He had been able to apply Schrödinger's equation to situations where the standard equation didn't work. He had not yet incorporated relativity. That was what was needed for a full blown theory of quantum electrodynamics (QED).

Almost all of the most important theoretical advances in fundamental physics in the late twentieth century were made possible by Feynman's reformulation because it made it possible to bracket out Heisenberg's uncertainty principle here and there. Measurement of a quantum system by a physicist causes "the collapse of the wave function." The observer inevitably eliminates the possibility of other measurements by making one measurement. Feynman figured out a way to look at the whole picture in a way that did not engage individual moments of a particle's position and thus didn't collapse the function, if I understand correctly.

Greatest Physics Genius of the 1900s

Last week I finished blogging through Thirty Years That Shook Physics, a great book by George Gamow on the first thirty some years of the 1900s, when the groundwork of modern physics was laid.

If I were to pick, I would pick Einstein as the dominant figure of the first twenty years of the twentieth century. Max Planck may have suggested the quantum, but Einstein ran with it and, in the meantime, established both special (1905) and general (1915) relativity.

For the next twenty some years, 1920-40, I pick Paul Dirac as the greatest mind in physics. There are lots of candidates. I pick Niels Bohr as the most influential figure, but I don't think the greatest mind. Heisenburg, Schrodinger came up with the most central concepts of quantum mechanics, but at times it seems like they just were the ones who did what someone else would have done anyway. It was Dirac who combined Schrodinger's famous wave equation with relativity, the last time that longed for synthesis was successfully accomplished.

But I'm not sure that any of those I've mentioned were as genius as Richard Feynman. I don't feel like I know enough to have a firm opinion yet, but Feynman may have been the most brilliant physicist of the modern era. The book I'm reading for the next few weeks for my Science Fridays is Lawrence Krauss' biography of Feynman, Quantum Man.

Introduction
Krauss begins with some fun memories of his own personal engagement with Feynman. A book given to him in high school about Feynman's notion that antiparticles were particles moving back in time (it was originally John Archibald Wheeler's idea). In college he had Feynman's lectures on hand. And of course he mentioned Feynman's moment in the spotlight, when he showed everyone why the Challenger blew up.  (He dropped an O ring into ice water on television)

1. Feynman in High School
Feynman was born in 1918. He was obviously a genius. Unlike Dirac, from whom you could hardly extract a word, Feynman was flamboyant and was loud. But he could also tunnel into a problem for long periods of time. He kept notebooks with multiple solutions to the same problems. he was methodical. He taught himself subjects years before he got to them in school.

He was incredibly gifted at math--not all physicists are, Einstein for instance. Of course you have to remember that when I say, "not gifted at math," I mean that Einstein didn't invent a new branch of math. Heisenburg reinvented matrix mechanics on his own. Dirac made major contributions to linear algebra for his relativistic quantum mechanics. So Feynman would invent path integrals. Meanwhile, Einstein couldn't have worked out general relativity without the help of David Hilbert.

One principle Feynman learned in high school that he didn't like but that would be key to his later fame was Fermat's principle of least time: light takes the path through various media that gets it most quickly to its ultimate destination. It is a weird concept because it seems to imply planning on the part of light, as if it knows where it wants to go and plans accordingly.

Apparently, another way to state this principle is this: an object will take that path where the total sum of the difference between its kinetic energy at each point and its potential energy at each point is lowest. This sum is called the "Lagrangian" and the "action" of an object.

2. Feynman at MIT and Princeton
Feynman declared as a physics major at the end of his first year at MIT. It was just right for him between the non-concrete pure math and the all concrete engineering. He and a student named Ted Welton started taking advanced graduate physics courses their sophomore year. One of those professors met with the two of them their junior year to teach them quantum mechanics. This was about 1938.

Then he turned down Harvard to go study at Princeton, which at that time had Einstein and John Archibald Wheeler. Feynman was Jewish, and his parents worried both about whether he could make a go at physics and whether he would be accepted as a Jew there. He and Wheeler (the guy that coined the phrase, "black hole") worked well together.

It was during this period that Wheeler suggested some particles might move back in time. The specific problem they tackled was the self-energy of an electron. They unsuccessfully tried to eliminate that factor from the energy equation because it went to infinity according to calculations at that time. Their suggestion was to try to eliminate the idea of a electromagnetic field. The idea was that interactions would only take place by the direct interaction of charged particles, by the exchange of photons. The prevailing wisdom was that forces interact by "action at a distance," which is what fields allegedly do.

Wheeler and Feynman were apparently wrong, but it was exactly the kind of interaction that fed Feynman's growing sense of possibilities.

Krauss takes a couple pages in this chapter to give a quick two points on the nature of quantum mechanics. The first point is that, in QM, objects can be in different places doing different things simultaneously. The second is the Heisenberg uncertainty principle. There are certain pairs of qualities (position and momentum, energy and time) where the more we know one, the less we know the other. The product of our uncertainty in knowing one and the uncertainty of knowing the other will never be smaller than Planck's constant.

That's probably enough for one day...

Quantum Paradigm Shifts

Scot McKnight drew attention to some fun waves in the field of quantum mechanics. Since I've been working through George Gamow's version of the early events, and some other books as well, this is a very timely article.

If this new approach is true, here is how the history of science will read. In the late 1920s, when the key discoveries of quantum mechanics were being formed, there were two basic perspectives in play. On the one side was Niels Bohr, a dominating personality who stood at the center of the burgeoning developments.

1. Bohr had somewhat of a cult following among the younger physicists. He brought them to Copenhagen and provided not only the most stimulating thinking on offer in physics but a thriving social setting as well. He offered the probabilistic interpretation of what was happening on the nuclear level.

The probabilistic interpretation is fun. For example, in this approach, the prevailing approach, an electron doesn't have a defined trajectory. It's as if it exists in a kind of indeterminate, undefined state. If you interact with it--say you try to define its position--you are the one calling it out of its indeterminate state. You may find a position but you have, in a sense, created it by trying to measure it.

2. But there were others in the mix who, in the crucial year 1927, resisted this interpretation. The exchanges between Einstein and Bohr are the stuff of legend. Einstein was a loner. He was the apple of the public's eye but Bohr was the attraction among actual physicists.

Einstein's refusal to go along with the trend is notorious and he is usually seen as a loser in his later years. When I teach about scientific revolutions in philosophy class, I've always used Einstein as an example of normal science resisting new paradigms. I always end with the line, "but Einstein's dead now and can't object." The point is that social factors often play a significant role in what becomes the prevailing scientific theory of the day.

But if this new approach is correct, then Einstein proves to be more correct than Bohr, who also is long gone and unable to object. He's not throwing any more physics parties in Copenhagen any more.

The center of attention in this discussion is de Broglie, who ironically started us down, inadvertantly, the path toward the probabilistic interpretation. He was the one who inspired Schrodinger's wave equation, which is the basis for the probabilistic interpretation. Einstein, Schrodinger, Dirac, they all did not fully accept the Copenhagen interpretation. But they lost the argument... at least till now.

3. The tremors in the force are coming from fluid mechanics. de Broglie originally suggested that "pilot waves" accompanied nuclear particles. In that sense, the fundamental constituents of matter were both particles and waves. This is slightly different from the Copenhagen interpretation, which sees matter, in a sense, as neither until we interact with it.

This article suggests that Richard Feymann, the most charismatic figure in the physics of the mid-twentieth century, may have been fundamentally wrong about his characteristically Copenhagenish interpretation of the double slit experiment. There seems to be a cosmic justice in that. I know he was a funny guy, but probably a little too cocky for his own britches. I feel the same way about Hawking, and I was delighted to see him bite the dust over the Higgs boson.

It will be interesting to see where, if anywhere, this whole line of interpretation goes. If there is a critical turn, we could see a ten year period in the near future that is almost as exciting as physics was in the late 1920s.