My theory was inspired by L. de Broglie and by brief but infinitely far-seeing remarks of A. Einstein (Berl. Ber. 1925, p. 9ff.) I was absolutely unaware of any genetic relationship with Heisenberg. I naturally knew about his theory, but because of the to me very difficult-appearing methods of transcendental algebra and the lack of Anschaulichkeit [visualizability], I felt deterred, by it, if not to say repelled. [1]Heisenberg responded in a letter to Pauli:
The more I think about the physical portion of the Schroedinger theory, the more repulsive [abscheulich] I find it....What Schroedinger writes about the visualizability of his theory 'is probably not quite right', in other words it's crap. [2]Most of the philosophical debates swirling around quantum mechanics have to do with causality. We all know what a wet blanket Einstein was on Las Vegas night. (Probably still is. "Put those dice DOWN! I'm talking to you, God!")
But in the childhood of quantum theory, the matter of visualizability loomed just as large. In his second paper on wave mechanics, Schroedinger wrote:
...it has even been doubted whether what goes on in an atom can be described within a scheme of space and time. From a philosophical standpoint, I should consider a conclusive decision in this sense as equivalent to a complete surrender. For we cannot really avoid our thinking in terms of space and time, and what we cannot comprehend within it, we cannot comprehend at all. There are such things but I do not believe that atomic structure is one of them. [3]Schroedinger wrote to Willy Wien:
Bohr's standpoint, that a space-time description is impossible, I reject a limine... If [atomic research] cannot be fitted into space and time, then it fails in its whole aim and one does not know what purpose it really serves. [4]Bohr and Heisenberg of course held a different opinion. The founding papers on matrix mechanics expressed the operational philosophy: "You got your equations, you got your observations, and they match. What more do you want? Shut up and calculate!" Of course, they had to say it more politely, at least in print. For example, here's the abstract, in full, of Heisenberg's famous paper:
The present paper seeks to establish a basis for theoretical quantum mechanics founded exclusively upon relationships between quantities which in principle are observable. [5]And in the introduction to the "Dreimaennerarbeit", the paper that laid out the whole structure for the first time:
Admittedly, such a system of quantum-theoretical relations between observable quantities...would labor under the disadvantage of not being directly amenable to a geometrically visualizable interpretation, since the motion of electrons cannot be described in terms of the familiar concepts of space and time. [6]but then they immediately point out what really counts: the equations of motion have the same form as in classical physics. Yet in the same paragraph, they concede:
In the further development of the theory, an important task will lie in the closer investigation of the nature of this correspondence and in the description of the manner in which symbolic quantum geometry goes over into visualizable classical geometry.Echoes of this argument reverberate faintly today. Textbooks complain about the desire for pictures (see Feynman vol.III pp. 11-4, 11-5 for an example). Pop-science articles glory in "quantum weirdness".
Philosophically, I can't say Schroedinger's position makes sense to me. Our visual systems sport some pretty complicated circuity, a lot of it quite non-intuitive. (Read Hubel's Eye, Brain, and Vision for the nifty details.) It's first rate for watching movies and playing video games, and similar Cro-Magnon activities. It didn't evolve to help us understand quantum field theory!
Historically, common sense intuitions have a pretty poor track record. Who invented statistical mechanics anyway, a masochist during a New England winter? "Brrr, it's cold outside! Hit me harder, molecules, oh, that feels gooood..."
On the other hand, I'm a visualizer myself. I started the "photons, schmotons" thread with one question in mind: just how far can you visualize the QFT description of light? I had in mind something like the balloon analogy in GR: it's wrong because of (a), (b), and (c), but it's still useful because it does capture (e) and (f). Little did I know...
Perhaps by the time the infamous, never-ending, hydra-headed "Photons, Schmotons" thread runs its course, my question will be answered. For the rest of this post though I want to talk about the discarded pictures (discarded by QFT, at any rate). Presumably deadly experimental results could be marshalled to drive stakes through the hearts of all these alternatives, but I'll leave that for someone else to discuss. Pretend we've turned up an old family album in the attic. Each quaint sepia-toned photograph draws our interest and affection. Someone else can recount how prosperous-looking Uncle Max went bankrupt in 1926.
OK, let's say we're determined to visualize wave-particle duality, experiments be hanged! What are our options? I can think of four.
The Ten-Minute History of Science says, "Newton, light particles---BAD! Huyghens, light waves---GOOD!" It comes as a bit of surprise to learn that Newton's Opticks is filled with observations of interference and diffraction phenomena. Newton concluded that his corpuscles had to undergo a periodic change of state, swinging back and forth between Fits of easy Reflection and Fits of easy Transmission.
Newton's theory of light had three characteristics:
How does Huyghens stack up?
...when a Ray of Light falls upon the Surface any pellucid Body, and is there refracted or reflected, may not Waves of Vibrations, or Tremors, be thereby excited... and are not these Vibrations propagated from the point of Incidence to great distances? And do they not overtake the Rays of Light, and by overtaking them successively, do they not put them into the Fits of easy Reflexion and easy Transmission described above? [9]
According to the pop-history of science, Planck's theory fell into this category. For example:
Imagine a sponge in a bathtub... According to Maxwell, when a sponge is squeezed it sends out its water in the the usual way and causes waves in the bathtub. Planck's sponge is of a rarer sort. Indeed it is more like a bunch of grapes than a sponge, consisting of myriads of tiny balloons of various sizes, each full of water. When this sponge is squeezed, the balloons burst one after the other, each shooting out its contents in a single quick explosion--- a bundle of water--- and setting up waves... Einstein, however, took the sponge right out of the bathtub... When he squeezed his sponge gently, water fell from it like shimmering drops of rain. [10]A charming story, but historically all wrong! Kuhn [11] argues persuasively that Planck believed in a completely continuous theory--- continuous waves, continuous emission and absorption--- until 1908, after Einstein put forward his light quantum hypothesis.
However, in 1912 Planck did come up with his so-called "second theory", in which emission is discontinuous, while propagation and absorption remain continuous.
Kuhn's book has full details. Though Planck's second theory never made it to the big time, it did come up with two hits: zero-point energy made its first appearance here, and Bohr got some inspiration for his model of the atom.
There are today not a few physicists who, exactly in the sense of Mach and Kirchhoff, see the task of physical theory to be merely the most economial description of empirical connections between observable quantities... In this view, mathematical equivalence means almost the same as physical equivalence. [12]So is matrix mechanics just as good as wave mechanics, or maybe even better, because it doesn't clutter up the story with fairy tales? No, say Schroedinger--- physicists need space-time (i.e., pictorial) descriptions to make progress. He then proposes an interpretation of the wavefunction psi: the real part of (psi d psi/dt) gives the spatial density of electric charge.
Schroedinger also constructed a wave-packet: a well-localized psi function that stays together in time. He did this for a harmonic oscillator potential (just our coherent states, I'll bet!), but he hoped originally to do the same in general.
All waves, no particles anywhere! Can it really be that simple? Schroedinger hoped so.
Schroedinger sent his papers to "grey eminence of theoretical physics", Hendrik Lorentz. (Lorentz incidentally was the first fellow to convince Planck that the black-body formula could not be derived without some sort of discontinuity assumption.)
Lorentz raised several objections [13]. First, he noted that psi is function of (x,y,z) only in the single-particle case. With two particles, psi becomes a function of six variables, the coordinates of both particles:
If I had to choose between wave mechanics and matrix mechanics, I would give preference to the former because of its greater Anschaulichkeit, so long as one is concerned only with the coordinates x,y,z. With a greater number of degrees of freedom, however, I cannot interpret physically the waves and vibrations in q-space and I must decide for matrix mechanics.Lorentz also pointed out that the harmonic oscillator potential was quite special, and that in the field of a hydrogen atom, the wave packet would spread out rapidly.
I won't go through the rest of Lorentz's criticisms. Schroedinger's biographer notes:
Lorentz belonged to an older generation of physicists, and Schroedinger might have drawn from their discussions the conclusion that his new discoveries cannot be fitted into a classical framework at all.So all these are wrong! But what's right? Stay tuned...
[1] Ann. Phys., v.79, 734-56; quoted and translated in Schroedinger: Life and Thought, by Walter Moore, CUP, 1989, p.211.
[2] Heisenberg to Pauli, 8 June 1926; quoted and translated in Uncertainty: The Life and Science of Werner Heisenberg, by David Cassidy, W.H. Freeman and Co., 1992, p.215.
[3] Moore, p.208.
[4] Moore, p.226.
[5] Heisenberg, "Quantum-Theoretical Re-interpretation of Kinematic and Mechanical Relations", in Sources of Quantum Mechanics, ed. B.L. van der Waerden, Dover, 1968.
[6] Born, Heisenberg, and Jordan, "On Quantum Mechanics II", in Sources of Quantum Mechanics.
[7] Or so says I. Bernard Cohen in the preface to the Dover edition of the Opticks (see page xlvii).
[8] Why not three out of three, if we believe in quantum mechanics? I side with I.Bernard Cohen: "...we must choose between (1) the historical or (2) the antiquarian approach to the development of science... the antiquarian's sifting of the disjecta membra of the Opticks (often out of context) for an occasional 'precursorship' of one or another 20th-century physical concept." (op. cit.)
[9] Newton, Opticks, III.1 Query 17. And yes, I'm being a bit antiquarian here.
[10] Banesh Hoffmann, The Strange Story of the Quantum, 2nd ed., Dover, 1959, p. 26.
[11] Kuhn, Black Body Theory and the Quantum Discontinuity, 1894-1912, OUP 1978.
[12] Moore, p.212.
[13] See the discussion in Moore, pp. 214-217. This is the source for the two quotes below.
Next:
Notational Conventions
Up:
Schmotons
Previous:
Michael Weiss: Flooding the Spacetime Plain
side! Hit me harder, molecules, oh, that feels gooood..."
On the other hand, I'm a visualizer myself. I started the "photons, schmotons" thread with one question in mind: just how far can you visualize the QFT description of light? I had in mind something like the balloon analogy in GR: it's wrong because of (a), (b), and (c), but it's still useful because it does capture (e) and (f). Little did I know...
Perhaps by the time the infamous, never-endingbaker.htm 0100644 0002041 0000013 00000024136 06503322064 0013523 0 ustar 00baez mathprof 0000441 0252514
But first--oh professor, I think you took too much off! You say:
This corrects a little mistake of Michael's where he claimed thatBut I explicitly said I was redefining the zero-point of energy to get rid of those pesky factors of e-it/2. Hmm, maybe you're saying the formulas are trying to tell me something-- that I shouldn't monkey around with the zero-point? (Ominous music wells up on the soundtrack). (Flash to the final scene in a 50s sci-fi movie, as the grey-haired senior scientist portentously intones, ``There are aspects of Nature that we change at our peril. Let this be a Lesson To Us All...'')e-itH Coh(z) = Coh(e-it z). It's not quite so simple and nice. Hint: vacuum energy.
(Actually, the grey-haired senior scientist has already said something about this zero-point stuff.)
Okay.
A) Use the formula
to get a curiously similar formula involving an exponential of only creation operators, applied to the vacuum.
Here z=c+ib.
I did almost this computation once before, but let's do a quick recap.
First: we want to show that Coh(z) is an eigenvector of the annihilation operator a. For this we need to compute some commutators, and the slick way is to notice that [a,-] acts like a derivative on many operators. At least this works for power series in a and a*. Say f(a,a*) is a power series with complex coefficients. Since [a,a]=0 and [a,a*]=1, we'll get the right result for [a,f(a,a*)] with this recipe: compute (d/dx)f(a,x) formally, treating a like a constant; then replace x with a* in the final result.
Using this rule on eza* - z*a, we get
Now put |0> on the right, we get
since a annihilates |0>. So a Coh(z) = z/sq2 Coh(z).
Next we expand Coh(z) in the basis |0>, |1> ....From the eigenvalue equation, we get immediately:
where C0 is the coefficient of |0>.
We can evaluate |C0| pretty easily. The norm squared of Coh(z)is |C0|2 exp(|z|2/2), from the formula we just got. But Coh(z) has norm 1. How do I know that? Well, e(za*-z*a)/ sq2 is unitary. How do I know that? Well, (za*-z*a)/ sq2 is i times a self-adjoint operator (just take the adjoint and see what you get), so by some theorem or other its exponential is unitary.
So |C0| = exp(-|z|2/4). So we've determined e(za*-z*a)/sq2 |0> up to a phase (let's call the phase iota):
Hmmm, now for a new twist. The professor asked for the answer in terms of a*. Well, a*n |0> = sqrt(n!) |n> --hey, this works out nicely:
What are we going to do about that phase iota?
Hmmm, let's take another approach. If life was really simple, we could just say that eza*-z*a = eza*e-z*a (it isn't), and since a annihilates |0>, e-z*a|0> = |0> (just expand out e-z*a in a Taylor series). So we'd have:
But skimming back over the thread, we get strong hints that
Let's ask. Oh, professor!
The Baker-Campbell-Hausdorff formula says that when [A,B] commutes with everythingHey, keen! How do I prove that?eA+B = e-[A,B]/2 eAeB
You don't. You thank Baker, Campbell, and Hausdorff for proving it.Okay, thanks! (They all read the newgroups? I've seen a post from Galileo, so maybe.)
Well, that makes short work of this half of the problem. Let's set:
which is a number and so commutes with everything, so
so
so the factor iota is 1.
Whew! Heavy firepower, just to determine that measly little phase factor iota! But then, rumor has it that Gauss spent two years of Sundays just trying to determine the sign of a certain square root.
Before I dig into the business of working out what coherent states look like in the basis of eigenstates of the harmonic oscillator Hamiltonian, let me comment on one thing:Uh-oh, better get the rest of my homework in before the professor goes over the assignment in class.
OK, last time I figured out that the coherent state you get by sliding the Gaussian bump c units to the right:
is proportional to this, in the basis of eigenstates of the Hamiltonian, aka ``particle representation'':
where D=c/ sq2 [sq2 being our symbol for sqrt(2)]. This time I've included the factor exp(-D2/2) so as to get a normalized state-vector. Since e-icp is a unitary operator, by some theorem or other, e-icp |0> is also normalized. So we've expressed the coherent state in the particle representation, up to a phase.
The probability distribution of this state-vector is a Poisson distribution with mean value D2:
(To do the mean-value sum, notice that the n=0 term politely disappears, and if you pull out a factor of D2, what's left is just the sum of all the probabilities-- which had better add up to 1.)
Now this is nice. The original question was:
``If an electromagnetic wave has amplitude A and angular frequency w, how many photons does it contain per unit volume, on the average?''and dimensional analysis said: A2/w. Well, presumably c and D will end up being proportional to A, once we actually start talking about electromagnetic waves, and not this mickey-mouse-dot racing around a circle! And the mean value should be proportional to the photon density. So we have our explanation for the A2 factor.
But to make this convincing, we have to know that D really does correspond to the radius of the dot's racetrack. So we need the time-evolution of the coherent state.
Now here I got stuck for a while, for I ignored the old adage: ``Never compute anything in physics unless you already know the answer!'' (Who said that, Wheeler?) I was trying to show that e-icp |0> would evolve to e-icp cos t|0>, since that's the formula for simple harmonic motion.
No good! The coherent state wavefunction has both position and momentum encoded in it. You can't expect the position to change without the momentum also changing.
OK, let's start over. First, we have to consider a more general coherent state, say one with momentum b and position c, thus:
which I'll call Coh1(c+ib). [ Coh1 because it's ``coherent state, take 1'': John Baez will shortly introduce a better choice. --ed.]
Now we want to express this in the particle representation, because we know how the states |n> evolve: |n> evolves to e-int |n> (if we factor out the common e-it/2, i.e., redefine the energy zero-point).
I did this for the special coherent state Coh1(c) a while back, but I was working entirely too hard. I used the fact:
Now this looks a lot like a derivative formula: [q,pn] = i (dpn/dp). (Ignore the fact that I haven't defined d/dp.) Actually, this shouldn't be surprising: we know that
i.e., bracketing with p is just about the same as taking derivatives with respect to x. Now q and x are pretty closely related, and what holds for q ought to hold for p, using Fourier transforms and all.
So we should have:
at least if f(q) is a power series in q, and g(p) is a power series in p.
As a check, let's verify the product rule. If that works, then we should have our result for all powers of p and q just by induction, and then for all power series by continuity arguments. The continuity arguments might take up a chapter or two in a functional analysis textbook, but hey, I'm sure the kindly moderator will cut us some slack. (Pause for thunderbolts to dissipate.)
It works!
So:
You have to be a little bit careful here. eA eB is not generally equal to eA+B if A and B don't commute.
If that last equation doesn't leap out at you from the previous two, well it's just a bit of straightforward grinding.
And so:
So Coh1(c+ib) is an eigenvector of q+ip with eigenvalue c+ib. It then follows that, up to a phase (call it iota):
Remember now that we want to time-evolve this. (So much physics, so little e-iHt...) As I said earlier, |n> evolves to e-int |n> (if choose our energy zero-point so as to get rid of the vacuum energy).
So the n-th term of our formula for Coh1(c+ib) will acquire the factor e-int in t-seconds. But we can absorb this into the factor (c+ib)n, just by replacing c+ib with (c+ib)e-it. So:
Roll over Beethoven, how classical can you get! If I told you that Coh1(c+ib) was my symbol for a dot in the complex plane at position c+ib, you'd say the equation I just wrote is obvious.
[Moderator's note: The aphorism, ``Never calculate anything until you know the answer,'' is indeed due to Wheeler. It appears in Taylor and Wheeler's book Spacetime Physics under the name of ``Wheeler's First Moral Principle.'' No other moral principles are mentioned, so maybe it's Wheeler's Only Moral Principle. --Ted Bunn]
In the last episode, Michael and John worked out the complex structure for the Hilbert space describing a single foton. We now eavesdrop on an email exchange between them!]
Michael Weiss writes:
I may be a little distracted from QFT for a bit.
At odd moments, though, like sitting in traffic, I find myself wondering "OK, that's the complex structure, now what's the inner-product? JB gave a formula to get the inner-product from the symplectic form. Well, what's the symplectic form?"
Are these the right things to be wondering about? Or should I concentrate on driving?
John replies:
In Boston, you should definitely concentrate on driving.
However, when you are sure nobody is about to plow into you, here are some things to ponder:
First, I've already told you the whole Hilbert space structure - in Fourier transform land, it's just L^2 of the forwards lightcone - so the question is really what we want to do with it.
Michael interjects:
What about the backwards lightcone? Seems a shame, after going to all that work to figure out a complex structure for *both* halves of the lightcone, just to throw away the bottom half.
Of course, for the subspace of real functions, the values on the bottom half are determined by the values on the top half. But I thought the idea was that our complex vector space of real-valued functions sits inside the complex vector space of complex-valued functions in a copacetic way. I assume this remains true when we add in the Hilbert space structure.
John concurs:
Right, that's true. If we're studying *real* solutions of the wave equation in the Fourier transform picture, we only need to worry about their values on the forwards light cone - since that determines their values on the backwards light cone. If we're studying *complex* solutions we need to keep track of both the forwards and backwards lightcone. In that case, the Hilbert space is L^2 of the forwards lightcone with its usual complex structure, direct summed with L^2 of the backwards lightcone with *minus* the usual complex structure.
Anyway, there are various things we could do.
One thing we could do is see what the Hilbert space of our "foton" looks like in terms of initial data - meaning the field u and its first time derivative udot at t = 0. The symplectic structure, the complex structure, the real inner product - the whole schmear! To figure this out, we need to do little Fourier transforming. Good exercise, that.
Michael responds:
Sounds like a good project. Is this going to help me understand "non-localization of fotons"?
John:
Hmm. I'm actually not an expert on the nonlocalizability of massless particles. To creep up on *that* issue, it would probably be better to first study *massive* particles, where the Newton-Wigner localization *does* make sense, and then see how it blows up in our face when we set the mass to zero.
The above project would be a good warmup exercise for all that stuff - it'd get our Fourier transform skills back up to shape. But the more immediate payoff of understanding the Hilbert space structure in terms of initial data is that we'd understand some of the funky-looking equations people write down when they talk about "canonical commutation relations" in quantum field theory. After all, the commutators come straight out of the symplectic structure!
Also, we would get ourselves into the position of being able to understand coherent states of fotons, which is a good warmup for understanding coherent states of actual photons. We know all there is to know about coherent states of the harmonic oscillator; once we understand exactly how the Hilbert space of a single foton works, we'll be able to put that knowledge to good use!
cinema.htm 0100644 0002041 0000013 00000005054 06503322402 0013665 0 ustar 00baez mathprof 0000441 0252514
Camera pans in on John Baez, strapped in a chair. A maniacal Michael
Weiss hovers over him, holding a dentist's drill, whose tip contains,
instead a diamond, a gleaming photon. Michael asks with a grating
voice that sends phonons racing up and down one's spine (simultaneously):
Baez groggily looks around. ``Huh. Most people just post to
sci.physics.research and hope for someone to answer, not kidnap
the moderator and strap him to a dentist's chair! I know I'm overdue
for a checkup, but this is ridiculous. Are you getting back at me for
avoiding questions about ontology, or something? Is what real,
anyway?''
``Shaddup, wiseguy.'' Weiss clobbers Baez with a cosh.
After a few more bizarre special effects, changes of scene, and
a whole lot of dreamy music, Baez wakes up. All he remembers is a
question...
So...let me go back to our general picture of coherent state.
What's the best quantum approximation to a classical particle on
the line with specified position and momentum? A Gaussian bump with a corkscrew twist!
We will only be interested here in bumps that have equal uncertainty
in position and momentum: The simplest case is when
Then we use the ground state of the harmonic oscillator:
where I left out the normalization factor to reduce clutter.
To get coherent states with other expectation values of position and
momentum, say
we can take our ground state, translate it in position space by an amount
c, and then translate it in momentum space by an amount b:
where I have attempted to make an even number of sign errors. [An allusion
to Dirac's comment on the first presentation of the Klein-Nishina formula
(by Nishina). See Gamow, Thirty Years That Shook Physics. --ed.]
But wait! We could also have translated it first in momentum
space and then in position space, getting
How does this answer fit with the other? Well, it differs only by a
phase.
``Only a phase''...ah, what an understatement! When physicists and mathematicians
mutter darkly about ``cocycles'', ``projective representations'', ``double
covers'', ``central extensions'', and even more intimidating things like
``anomalies'', ``the Virasoro algebra'' and ``affine Lie algebras'', they
are secretly complaining about the many subtleties that can caused by a
mere phase!
So let us think about this a little bit. The two coherent states above
differ by the phase e-ibc. That should be no surprise;
the Heisenberg commutations relations
lead directly -- with a dose of mathematical optimism -- to the exponentiated
version called the ``Weyl commutation relations''
which describe how translations in position space and momentum space
commute only up to a phase. Actually, mathematical physicists of the rigorous
variety prefer to take the Weyl relations as basic and derive the Heisenberg
relations as a consequence. But we are being relaxed here so we can think
of them as two ways of saying the same thing.
Now, Michael has taken
as his definition of a coherent state with expected momentum b
and expected position c. This is fine...up to a phase...but it's slightly
annoying how one needs to ``break the symmetry'' between momentum and position
in this definition. Why not
Or even better, how about some choice that treats position and momentum
even-handedly! ``Mind your p's and q's!'' There's much wisdom
in that phrase...
Here's a nice way to mind our p's and q's; we make the
following new definition:
Here we ``simultaneously translate in position and momentum space''
instead of favoring one or the other. This state is not equal to either
of the two choices listed above, but again it differs only by a phase.
Why?
Well, one can show that
at least if I've not made a sign error. So this new definition ``steers
a middle course'' between the other two choices, phase-wise.
(Also, fans of symplectic geometry will appreciate the funny skew- symmetric
quality of the expression -icp + ibq in our new definition.
But let's not get into that.)
Here's a little assignment for Michael, or any other students willing
to pitch in! Remember that c represents position and b
above represents momentum, so (c,b) represents a point
in phase space. Also remember that it's good to think of phase space here
as the complex plane. So let's define
(Don't confuse this lower-case z with the upper-case Z we
had before. The upper-case Z was an exhalted operator; the lower-case
z is just a lowly complex number.)
Now: take the expression
and rewrite it in terms of z and its complex conjugate z*,
while simultaneously rewriting p and q in terms of annihilation
and creation operators. Remember that
Some nice stuff should happen.
If we do this, we will get a cool expression for our coherent state
in terms of annihilation and creation operators applied to the vacuum state.
This won't immediately solve all our problems, but it should help us understand
a lot about how our coherent states of the harmonic oscillator evolve
in time.
ܦS^rpS*q0bٸEE4%&R5* 2E$<J
jIF)ef$ V$D Z슉 6T:H:}cR<
2 4r(do.!`~/'lWbI^*k$tcMN)r(J
MOsN艚
F + ݲ"wxFk+\I])9L.-x G#0VX4g m:i<BDR,r1U'a((IԛtѴR kLQ5
*9d &G!\3!K( ҨB\Wvqz0Iic21_
r
The diagram stretches out to infinity in all directions, of course,
but let's say this piece just goes from w=-1 to w=+1, and k=-1 to
k=+1. At the four corners we *could* put the four functions:
which span a 4-complex-dimensional subspace of the space of *all*
complex-valued functions on R. (Nice thing about this post - I can
just talk about complex vector spaces, and fuggedabout inner-products,
measures, and so forth for the nonce.)
Here I'm using the *wrong* complex structure - the one where to
compute i times f(x), you just multiply using ordinary complex
arithmetic. With that definition, I'm sure our four functions are
linearly independent.
With John's *clever* complex structure, I get a little confused. Do
we really have four independent vectors, or am I seeing double?
Let me think about the *real-valued* functions first. I've been
working with these four till now, two that travel rightward, two
leftward:
Since these are all real-valued, we can form a real vector space with
them, no sweat. 4-dimensional as a real vector, natch.
Where do these go on my "Rule Britannia" diagram? Well, =
that's a bit
ambiguous. Does cos(x - t) has positive frequency, or should I think
of it as cos(-x + t) with negative frequency? Both and neither!
Blame Euler, who noticed that cos(x - t) is the average of *two* of
our exp waves, the positive frequency exp(i(x - t)) and the negative
frequency exp(i(-x +t)). We have the problem deciding whether
sin(x - t) has positive or negative frequency.
Anyway, following John's instructions, we now define the operation of
J on sin(x - t) by the simple rule: "roll back 1/4 period".
This gives cos(x - t), either way you look at it. Likewise J applied
to cos(x - t) gives -sin(x - t) (as it must, if we're gonna have
J^2 = -1).
And that turns our 4-dimensional *real* vector space into a 2-dimensional
*complex* vector
space! After all, if you know what r times v means, for any real
number r and vector v, *and* you know what i times v means, then you
*ought* to know what (r+is) times v means--- or, to be crystal clear,
(r+Js) times v. Our two rightward waves now become linearly
dependent; one's just J times the other.
How about our exp waves? Hmm, Euler always said, "shut up and
calculate". (Or something like that.) Let's try this:
Seems a little odd. But OK, I guess. I suppose I'm trying to find a
complex-linear map from:
This map has a couple of nice properties. First, say f belongs to the
first space, but just happens to be real-valued. Then the map leaves
f alone - it just maps f to itself. Next, the *negative* frequency
waves just get annihilated! How's *that* for the power of positive
thinking!
(Hmm, you think I can throw away my glasses?)
Next:
Next: John Baez: A Rather
Up: Schmotons
Previous: Michael Weiss: A Trivial
Cinematic Interlude
``Is it real?''
Michael Weiss
3/10/1998
circle.gif 0100644 0002041 0000013 00000010365 10207501346 0013652 0 ustar 00baez mathprof 0000441 0252514 GIF89a f f3 3f3ff̙f f 3 ff3f̙3 ff3 w f U 33333ffff33fff3f3ff33ff̙ 3f 3 33f3fff ̙ f3̙ f3ff̙3f 3 w f U 3 3333ffff3f3ff̙3 3 f3f 3f ff 3f ff 33 fff3̙̙3 ff̙wwwfffUUU333 ! p , p=G?p9FD.DU6H=+DNƇ8ͱ/O;ՙ8ER݅*W͡/LLF2/p p "V AF7j ޢ7ƓPh5:AIs " PC@H2Q7 t |(AӫPbґ5Ek.طZ 7e x_T*Ut
^*6" m} bO9HS捍ˠI0CШ7 9c[UцϲsSZCoO+ ē[JQD\ zTZN(FA/>_8 K>5\F(`r G qL
Next: Michael
Weiss: Now for Up: Schmotons Previous: Michael
Weiss: Roll over
John Baez: A Gaussian bump with
a corkscrew twist!
OK, last time I figured out that the coherent state you get
by sliding the Gaussian bump c units to the right ...
Very nice. I would like to understand this better and think more about
the best way of deriving it. The way I suggested to you was stupid and
grungy, but you seem to have fought your way through and then discovered
some much nicer approaches. I haven't thought about this stuff enough,
so I'd like to polish your work to a fine sheen before moving on.
Next: Michael
Weiss: Now for Up: Schmotons Previous: Michael
Weiss: Roll over
Michael Weiss
3/10/1998
corkscrw.old.gif 0100644 0002041 0000013 00000420453 06503521164 0015032 0 ustar 00baez mathprof 0000441 0252514 GIF87a? ̻wwwfffUUUDDD333"""f3 ̙f3 f3 ffffff3f 3333f333 f 3f3 ̙f3 ̙̙̙̙f̙3̙ ffffff3f 3333f333 f 3̙f3 ̙̙f3 ̙f3 ff̙ffff3f 33̙33f333 ̙ f 3ffffff3f fff̙fff3f ffffff3f ffffffff3ff f3f3f3f3ff33f3 f f f f ff 33333f333 333̙3f333 3333f333 3f3f3f3ff3f33f 33333333f33 3 3 3 3 f3 3 f 3 ̙ f 3 f w f U D 3 " w f U D 3 " w f U D 3 " , ? ! HB0p 6 8b d F!FLqȃ%G"L9"ʊ3ˏ !hQ&@lE1&ŞH iC:EtԄ:2+R^ILʕk¨Ѧ%ИF;mܴF;נAʲY&)ӁFB>F?ْrGY͕Mn~`OƯ'Si0LmPm@=7lIqS:
.ԃ[2Uտ _eλb^2S3#<ßha}E5muS(O7xAJ̈>-*hJ
ǩ0:SR@1nc*/jC
i)` YW5蔂O*9y,aIγ7u?M$'Tp =UVZRŽKX;H1IJ0C:"^`CIo-|: *O:.p#p۳hnB1~ݼuJzSӈy*C>l kb]0'Qk{ 7ݹ]" c{^]M{O; m=9o{͊|xʽ KɦJ/IC1JR}c*i늡@2z9̩$Y|ޫ1Uy \R1tqʡDrX/b,mU, &=cJ(-GAi ʻWdkh ڤ'\ $ZhuEl*Z Q1+UswRDR-V]
B!@9%E$"pOX&
u4ULMg2Eu>&Ēuwa|O5*Pg_.-0x əD$D
"$l +Kj熂ԥi ziTDf|E N,.# ;`KN,[bЄOT{j)jL@$̅ZֱNCvGtf`
br.M!WOtwG]S& 0}g4 - Ϝt4BhR133{U4ٓVcutKީ3\v'&!Ob^.9_%RړѧŘōӵ Eڍqbrjݖ=}iz^`,2".,!u"UzʵIL_0(gnQ4xP$bfi&HТ̂."'f44"ҴoLbNNeމr*j4H~<)B
D&"G(Σ}>V*ƢH8Ji~
)12j3'i8S#t%8)E#w fgfpZ( QDvNܙ`QHs$R=UCdhIsO_5ZaM' DKep}S:,SS3_C*Ab%0T G=U/k)E&*"XgAIv2E[b}37<$sYed &[+hXVXH6W~$Wz0S-kȚ|$T8GO'203:@+C['ѤB|1d&:MTN4,0kLMEJ$a$aB3tV'krxgq #z+5:"P+OzƘ` M%Tfԩ))qQ+ihѣ`O]qcV$,6W$] ǞguA#F&-Sf?t9/sn'X$=gfdc%S ר4S_q0 Z=y!Q`PAI 3P僿 O xE>F a()qt:Q
젟 ͢u=q۾W]tKM[d>lvJc:2!AYمZYKbE:b"KYm 7;v9b'FKVzt YHkY,٤F{qUb_LbNa՞7
&[$ɭ~Oע(=ܢkZ&v3g(` ~kx6FnpF8\=4K-a|C&Ktc{st%6TDnl#'n8фMfomAӈ}( ]J ',V$=#%t?/7J& t30! 1%,-ju2
,=4-`b =NtAa1*9l@{XFP)8+{<71&_ɴ=]}ѵՈ3" cѡA`)S SB
q)r5 "t
#aW-ȫvh8*8iځiF<⹅7@R" V?c7|jpz 5G|(fAȩXt,"FwҖh>xY喍,Zv肙?V£ c1^קSFj|'{ө'e,>BU,IzO.;2SbR/ žrY`I/YqzG8`+PN^ES=v i `LJe8jtv3P-Q4)d;3 )D xA;Ql.[NѦ*EY\C0w^U:SCݷ_1ܚ-USߧ3 oHDif&8D1s!25O-`+&{Г$9[B>ok>qiP
+&cMts
V/V |=]/m8
u`m,]tp@c$][_̼:ۄF?u15d\'ұ(rkoa=_Qt9$
Next: Michael Weiss: Seeing Double
OK, John gave us the diagram:
\ /
\ / <------ forwards lightcone (w >= 0)
\/
/\
/ \ <------ backwards lightcone (w <= 0)
/ \
where pure waves exp(i(kx - wt)) live. Let me draw in the w and k
axes:
w
\ | /
\ | /
\|/
---------k
/|\
/ | \
/ | \
("Rule, Britannia! Britannia rules the waves"
- which I guess she did when Maxwell was alive...)
exp(i(-x -t)) exp(i(x - t))
exp(i(-x +t)) exp(i(x + t))
leftward rightward
--------- --------
sin(-x -t) sin(x - t)
cos(-x -t) cos(x - t)
exp( i(x - t)) = cos(x - t) + J sin(x - t) = 2 cos(x - t)
exp(-i(x - t)) = cos(x - t) - J sin (x - t) = 0
the space of complex-valued functions on R, with the ordinary
unclever boring "i" complex structure
to:
the space of real-valued functions on R, with the clever
interesting "J" complex structure
and I've found one! And it maps the 4-complex-dimensional space
spanned by the four corners of the British Empire, down to the
2-complex-dimensional space spanned by our trig waves.
Next: