Measurement of Action and the New
Imagine a perfectly transparent block sliding over a perfectly flat, smooth, and frictionless floor along a straight line that we identify as the x-axis of the coordinate grid that we superimpose upon the space in which we have set up this imaginary experiment. Initially the block slides in the positive x-direction, from left to right. Two extremely narrow beams of light oriented to travel in the direction parallel to the y-axis (the direction toward and away from us) hover just above the floor, each beam consisting of two beams of equal intensity propagating in opposite directions (so that the beams don't exert any net forces upon the block). A minuscule wedge of reflective material sits embedded within the block at the block's center of mass, so positioned and oriented that when the block passes through each of the beams the speck will reflect a brief pulse of light in the z-direction (up and down) to one of the detectors that we have hung above the floor. Carefully synchronized clocks in the detectors record when the pulses arrive, thereby giving us the data from which we calculate the time that elapses between the two reflection events that attend the block's passing through the two light beams.
After passing through the second beam, the block slides up a frictionless ramp, pauses at some point on the ramp, and then slides back down the ramp, back through the light beams, and toward its origin. Some suitable apparatus (perhaps merely a camera with its shutter held open) records where in the ramp the block came to a full stop. From the block's mass, the maximum height that the block's center of mass reached above the floor, and the acceleration of gravity in the z-direction we can calculate the maximum potential energy that the block gained when it went up the ramp. We then equate that energy to the kinetic energy that the block possessed while it slid across the floor.
Multiply that energy by the time that elapsed between the block's interrupting the two light beams and you will know the amount of action that accrued to the block's account between those two events. According to the quantum theorem, that amount must also include a term that represents the indeterminacies in the times and energies that we measure. But now we believe that we can see a way to eliminate that term and gain perfect knowledge of the events and of the block's dynamic properties, thereby invalidating the quantum theorem.
We need to bear in mind the fact that, as with Relativity, our intuitive understanding of Reality does not apply to the realm that we have under consideration here and that all of our efforts to make it do so will necessarily fail. We must rely entirely upon our logic, especially deductive logic, to guide us. Thus we know that the quantum theorem, which we deduced from the Principle of Least Action, tells us, by way of Heisenberg's indeterminacy principle, that we cannot measure both factors in an action with arbitrary precision. In the present case, to take the obvious example, if we measure the time interval with perfect precision, we cannot know the energy of the block with any precision at all. If we try to imagine how Reality enforces that rule, we will fail because our brains (and the minds that inhabit them) evolved to cope with existence at a certain scale: to understand Reality at scales different from that one we must replace intuition with reason. To paraphrase Francis Bacon, we must not imagine or suppose how Reality is shaped or acts, but we must discover the correct shape, composition, and behavior of the Existent. We have, in fact, conceived our imaginary experiment at the wrong scale, though we can still get away with it if we take proper care in our analysis (as we do with Relativity).
Our block and its frictionless floor and ramp give us the imagery of a Newtonian dynamical system. But if we want to look at actions on the order of the size of Planck's constant, we should more properly imagine an electrically charged particle drifting through empty space and then passing through a small hole in one of a pair of electrically charged flat plates, whose internal electric field stops the particle and throws it back through the hole. However, we can still use the block-and-floor model, just as we use ordinary trains and railroads in our derivations of the rules of Relativity, so long as we properly employ the applicable laws of physics in our imaginary experiments.
As I noted, if we measure the time interval between the block's beam interruptions with perfect precision, we cannot know the block's energy with any precision at all. Likewise, if we measure the block's energy with perfect precision, we will not be able to determine the time intervals with any precision at all. We have to think of the action as a kind of blurry picture: as we bring one part of the picture into sharper focus, we make some other part of the picture fuzzier.
To see what that means, look again at our imaginary experiment. On the first leg of the block's traverse we measure the time interval between the beam interruptions with almost perfect precision. We thus ensure that we have only the fuzziest idea of how much energy the block carried. Next we halt the block on the ramp and so prepare it that we have almost perfect knowledge of its potential energy relative to the floor. When we release the block onto the second leg of its traverse we know that we can obtain only the haziest notion of how time elapses for the block between the two beam interruptions. The situation plays out as if the block had expanded into a kind of pulsating fogbank (an image that shows us no more absurdity than does that of the shrinking trains of Relativity).
If we can determine that the block accrued the same amount of action on each leg of its traverse, we can invoke conservation of energy to assert that it carried precisely the same energy on both legs of the traverse. Then we can multiply the time interval measured on the first leg by the energy measured on the second leg, thereby obtaining a description of the action accrued on one leg or the other and its component variables with arbitrary precision. But doing that successfully would put us in violation of the quantum theorem. We have taken as our fundamental premise in drafting the Map of Physics the proposition that Reality so shapes itself as to uphold the most fundamental laws of physics, what I have called the Constitution of the Universe, and any laws that stand upon them. We have deduced the quantum theorem from those laws, so now we must figure out how Reality upholds the quantum theorem against our efforts to obtain perfectly precise descriptions of the block and its actions. More to the point, we must find mathematical descriptions of the block and its actions that automatically encode that upholding of the quantum theorem.
Through long experience we have become accustomed to the idea that in simple multiplication the order in which we write the multipliers does not affect the product: we say that multiplication of numbers obeys a commutative law. But not all multiplications obey the commutative law. The vector cross product, for example, does not: for vectors A and B we have AxB = -BxA. We describe electric and magnetic fields with vectors and we calculate the Poynting vector as their cross product, so we know that we have some examples in which our description of Reality requires non-commutative multiplication. Perhaps that fact can solve our problem?
Let's make a rule that when
we write down the variables that we measure in our quantum experiments, the
first variable that we measure goes on the right side of any product with
another variable and the second variable that we measure goes on the left side.
Thus we express the action that our block accrued on the first leg of its
traverse as Et and the action that it accrued on the second leg as tE. We can
then express the assumption that led to our violation of the quantum theorem as
And now we see how we went wrong.
Imagine that we follow a
particle from event A to event B and calculate the action that accrues to it. In
the same experiment we follow the particle from event A to event C and calculate
the action that accrues to it. In accordance with the quantum theorem, the
actions that we calculate between those pairs of events must include an
indeterminacy of one Planck unit. If we subtract the first of those actions from
the second we thereby calculate the action that accrues to the particle as it
goes from event B to event C. Naively, we expect the indeterminacies to cancel
out in the subtraction. But if we had determined the action between events B and
C directly, it would have to carry one Planck unit of indeterminacy, so our
subtraction must leave one Planck unit in order to conform to the requirements
of the quantum theorem. So now we know that Reality must have such a structure
that no subtraction of actions can eliminate that unit of indeterminacy from the
difference, which means that we must rewrite Equation 1 as
That analysis also applies to
actions that we calculate from the measurements of other variables. If we
measure the particle's location and its linear momentum, then we must have
true to Reality. And if we measure the
particle's angular displacement and its angular momentum, then we must have
true to Reality.
Equations 2, 3, and 4 tell us that when we describe actions with magnitudes not vastly greater than Planck's constant we must represent the things that we actually measure with mathematical entities that do not commute under multiplication. I have already mentioned vectors as non-commuting variables, but they don't give us the flexibility that we need to choose the magnitude of the difference between the alternate products. We get better possibilities if we use the two-dimensional analogues of vectors. In that case we would obtain the matrix mechanics version of the quantum theory that originated with Werner Heisenberg. I don't have a firm grasp of the group theory that matrix mechanics requires of its students, so I will leave to others the task of developing that version of the quantum theory for the Map of Physics. We get another possibility if we represent at least some of our measurable quantities with differential operators. In that case we will get the wave mechanics version of the quantum theory, the one that originated in the work of Louis de Broglie and Erwin Schrödinger. Since I originally learned quantum mechanics in the Schrödinger formulation, I will pursue that version in what follows.
and apply it to Equation 2. Of course the
differentiation operator cannot exist by itself: we must apply it to some
differentiable function, so we must assert the existence of a function Ψ= Ψ(t)
and rewrite Equation 2 as
Now we know that we must conduct our experiments, both real and imaginary, with the aim of determining the form and content of the state function Ψ.
In our sliding block experiment we measured the time interval between the defining events so precisely that we lost all possibility of measuring the block's energy with any precision at all. When the block comes to a stop on the ramp, then, its center of mass must do so over a range of heights above the floor. That doesn't mean that the block spreads out; that would not spread out the center of mass. We can only properly understand the spread of the center of mass as representing a probability distribution of the block; that is, for one instant the block's center of mass has only a probability of existing at any certain point in the range of possible locations on the ramp that we infer from Heisenberg's indeterminacy principle. In that circumstance how can we assign an energy to the block? We have to use a kind of average.
We assign to the block a
function (represented by ρ)
that represents a probability density; that is, at every point in space we
assign a probability per unit volume that the block's center of mass occupies
that point. If we multiply that function by the minuscule element of volume and
integrate the result over all space, we must obtain, of course,
which encodes the statement that the block
exists definitely somewhere. If the block has some property, such as energy,
that we can describe with some function of the block's location, then we can
calculate an average value for that property by multiplying the function that
represents it by the probability density function at the same point, multiplying
that product by the minuscule element of volume, and integrating. We get
which we call the expectation value of the energy because it tells us the most likely energy that we can expect the block to possess.
Having introduced two new functions, Ψ
into our discussion, we now want to determine the relationship that must exist
between them. Whatever form that relationship takes it must conform to the
requirement that ρ
represent only positive, real numbers (neither negative nor imaginary
probability has any physical meaning). We know that we cannot have ρ=Ψ,
because Equation 5 gives us
which requires either that Ψ increase or decrease continuously or that it represent a complex number. The first possibility violates the criterion encoded in Equation 7 and the second violates the requirement that we use only positive real numbers as probability densities.
We might next guess that ρ=Ψ2.
Because squaring a number always yields a positive real number, that expression
gives us a result that conforms to the requirement that the number representing
the probability density be both positive and real, regardless of whether the
numbers in the state function are negative and/or imaginary. But if we solve
Equation 9 for the state function, we get
(in which A represents an undetermined
amplitude) and we still have a problem: the square of that function still
violates Equation 7. However, if we make
then the square of Ψ
becomes a constant in time, which does not necessarily violate Equation 7, and
Equation 9 remains valid if we rewrite Equation 5 as
Thus we have the new quantum
theory that came out of the discoveries of 1924 to augment the old quantum
theory of Planck, Bohr, and Einstein. In addition to the energy operator in
Equation 12, we also have the differential operators representing linear
and angular momentum,
We can even combine operators to create new
operators, such as kinetic energy,
In the absence of potential energy (that is, in
field-free space) we can equate a body's kinetic energy to its total energy to
which is Schrödinger's equation. Now we need to
see what consequences this theory puts into our picture of Reality.
Appendix 1: Recap of the Logic
Given action conjugates expressed in generalized coordinates of linear momentum p and location q such that Heisenberg's indeterminacy relation applies, we infer that:
1. to preserve that relation in spite of any subtractions of actions we must have the commutation relation true to Reality;
2. to describe action with non-commuting multiplication we must use either matrices or differential operators to represent measurable quantities. We choose to use differential operators (due to familiarity), so we have .
3. use of differential operators necessitates the existence of a state function upon which they can act. The state function encodes a complete description of the system under consideration in order to accommodate any possible measurement.
4. differentiation of the state function is the
same as multiplying it by a number. If we calculate the action from
measurements, we must have S=pq+h. But we also have
We must be able to match those two equations, so
we must be able to divide the second of them evenly by Ψ.
So we have
divide that equation by Ψ
and h, multiply it by dq, and integrate it, we get
5. indeterminacy requires that measurable
quantities spread out. Only a probability distribution can achieve that goal in
a physical description, so we must associate with our system a probability
subject to the restriction that
6. Ψ and ρ both describe the state of the system and both are functions of p and q, so we must have a mathematical relation (ρ=ρ(Ψ)) between them, subject to the restrictions:
a. ρ is a positive and real number,
c. Ψ is an exponential function of the generalized coordinates.
So we infer that
i. Substitution of into the expression for ρ must yield pρ; therefore, ρ=Ψn.
ii. ρ cannot blow up or collapse over any span of q; therefore, Ψ must be an oscillatory function, so its argument must be imaginary.
iii. ρ must be a real and positive number; therefore the exponent n on Ψ must be an even number.
iv. n=2, because higher powers only multiply the argument of the exponential and thus give wrong magnitudes to the measurable quantities encoded within the state function.
7. the expectation value for the generalized
momentum of a particle comes from
in which the asterisk represents the fact that
we take the complex conjugate of the first psi when we carry out the
calculation, in accordance with the conventional way of squaring complex
Appendix 2: A Comment on
A. We do not actually measure energy or momentum. We can only measure distance and duration. From those measurements, properly contrived, we can infer momentum and energy and calculate numbers to represent them. For example, a ball rolling up a ramp shows us its energy when we see where it stops. We get that energy from a chain of inferences and experiments that give us the ball's mass, the acceleration of gravity, etc.
B. We describe the motions of bodies with curves (more precisely, with the equations that represent curves) and use measurements (boundary conditions) to determine the specific curve a body follows. The equations of motion reflect the environment (forcefields, constraints, etc.) and our measurements select out a specific possibility from all of the potential paths.
C. We associate with the
motion of a body over some path an action that we calculate from the body's
momentum and the distance the body traverses. We can measure distance first and
then momentum (via the body's climb up a ramp) or we can measure the momentum
first (by bouncing the body off an identical body that then goes up a ramp) and
then the distance traversed. In classical Newtonian physics the difference
between those two actions must equal zero. In quantum physics the difference
must equal Planck's constant. We might think that we could measure once to get
the momentum with δp=0,
repeat the experiment to measure the distance with δq=0,
and then, on the assumption that the two actions are perfectly equal to each
other, calculate the action with
δS=0. But Reality does not conform to that
thought. No matter how carefully we try to contrive the two actions to be equal,
we will always get the commutation relation
fact preserves the old quantum theory, which we deduced from the most
fundamental laws of Reality.
Back to Contents