This is a transcription of some notes I made after reading Quantum Mechanics: Symbolism of Atomic Measurements by Nobel Laureate Julian Schwinger, which I picked up after I heard Rui Vilela Mendes give a talk about the book’s ideas.
If you give me some time, this will be a part of a larger story about Schrodinger’s Aibo, about which I’ll say no more now.
Let’s first look at something that’s really weird to measure: the polarity of light.
You can think of a polarizer as a filter which has a very fine grid of parallel lines. If we orient this vertically (up-down), and we filter ordinary, white light, we get out all the light that was oriented (polarized) in an up-down direction. Turns out, this is about half the light we had. We can tell this from the relative intensity of the light that comes through the filter.
Now let’s take another polarizer, and stack it on the first one, but with the second polarizer’s grid oriented horizontally (90 degrees to the first polarizer’s orientation). We can say the new polarizer is filtering “left-right” polarization. When put together this way no light gets through the two sequential filters. This implies that up-down and left-right are mutually exclusive: all photons have one and only one of the two types of polarization. Filtering for both returns nothing.
Now let’s take another polarizer, and put it between the first two, but at a 45 degree angle to both of them. The weird thing is this: after inserting the third polarizer 12.5% of the original, white light gets through! So adding a filter increased the amount of light that got through!
It seems that polarizers do more than simply filter: they actively change something about the photons they are filtering.
This would be hell for a computer scientist: imagine a database where queries (filters of the objects in the database) affect the values in the database, via some internal program to which you have no real access (except through examining the results of sequences of queries). How could you figure out what (in general) you could expect from a series of queries?!?
To figure out what we might be able to do in this situation, let’s say we have a set of objects U, and a set of qualities that each object can have, Q. Q contains qualities like “A=address,” “B=boldness,” “C=company,” etc. Each quality can take on a number of possible values (A={a1,a2,a3…an). Let’s restrict our discussion to qualities where each object has to have one and only one value for any given quality. There are no empty data fields (or we could call “null” a particular value for a given quality). We call these complete qualities.
Let’s setup a query syntax G(U,A,a1), which means “give me all the objects in database U that have quality A set to value a1.” We can sum the results of queries to get an “OR” function, or perform queries in sequence to get a “AND” function. Let’s say that U is also a quality, which can take only one value u. This is a “universal” quality for which all objects have the same value. It helps later.
In normal databases, separate qualities are compatible: querying one has no effect on the values of another. In particular
G(G(G(U,A,a1),B,b1)A,a1)=G(G(U,A,a1),B,b1). That is, querying twice for a1 is redundant, and the fact we queried for b1 in between the a1 queries doesn’t change that. That means that the sequence of queries has no effect on the results: G(G(U,A,a1),B,b1)=G(G(U,B,b1),A,a1).
But what if this wasn’t the case? What if there was an unknowable machine operating, which changed the values of qualities in a database in a systematic but unfathomable way anytime you did a query? This seems to be the case with qualities like “up-down” and “left-right” polarizations. What can we say about such a situation?
Let’s try to describe what the unknowable machine does. To do this, let’s have a syntax for setting the value of a quantity: S(U,A,a1) sets the quality A for all the objects in U to value a1. Let’s also define a “fractional get”: a function that gets a fraction x of all the objects that have quality A set to value a1: F(U,A,a1,x). We’ll do lots of fractional gets and sets in sequence, so let’s say that S(F(U,A,a1,x),B,b1) ≡ SF(U,A,a1,x,B,b1).
What we need is some general properties for the fractions x(a1,b1) that result from sets and gets on A=a1 and B=b1, and other combinations of qualities and their values. Such properties are as much as we can hope to know about the unknowable machine.
To find these properties, let’s ask a question: for what function x(b,a) (that must be defined over all possible values for B and A) is the following statement true: G(G(U,A,a),B,b)=SF(U,A,a,x(b,a),B,b).
It may not be obvious, but for regular databases with compatible qualities, x(b,a) must equal 1 when a=b, and must equal 0 in all other cases. We will call this delta(b,a), the Kronecker delta function.
While x must be the delta function for compatible qualities, this isn’t the case for incompatible qualities. The question is: what properties (if any) must x(b,a) conform to in the incompatible case?
For compatible qualities, the following must be true, via reasoning about the delta function:
SF(SF(U,A,a,x(b,a),B,b),C,c,x(d,c),D,d)=F(SF(U,A,a,x(d,a),D,d),U,u,delta(c,b))
Let’s substitute x for delta, since that makes sense for compatible qualities:
SF(SF(U,A,a,x(b,a),B,b),C,c,x(d,c),D,d)=F(SF(U,A,a,x(d,a),D,d),U,u,x(c,b))
Since C and D are complete qualities, we know that if we sum over all their values, we have to get rid of those qualities:
Sum_C(Sum_D(F(F(SF(U,C,c,x(d,c),D,d),U,u,x(b,d)),U,u,x(c,a)))=SF(U,A,a,x(b,a),B,b)
For this to be possible, x() must have the following property:
x(a,c)=Sum_B(x(a,b) x(b,c))
The interesting thing about this is that if function x conforms to this property, it allows us to transform the effects of a query for A on B to the effects of a query for A on C. In other words, these functions relate all the various effects of the unknowable machine on various qualities to one another. For this reason, they are called transformation functions. Pretty important functions to know about.
It’s tempting to think that transformation functions are probabilities, since probabilities are the numbers we usually use to describe unknowable mechanics. But transformation functions aren’t probabilities.
Let’s say we have a function we can apply quality values called lambda(a), which returns a number, and that this number uniquely identifies the quality a. So we can invert lambda, such that lambda_invert(lambda(a)) gives us back a for all qualities and their values. It’s clear that for any such function:
SF(U,A,a,x(b,a),B,b)=F(F(SF(U,A,a, lambda(b)*x(b,a)*lambda_invert(a),B,b),U,u,lambda_invert(b)),U,u,lambda(a))
If the x’s were probabilities, all lambda(b)*x(b,a)*lambda_invert(a) would have to act like probabilities (stay between 0 and 1, and sum over all their values to yield 1). This can’t be the case for all invertible lambda, so the x’s can’t be probabilities.
What they can be is the square roots of conditional probabilities: p(b|a)=x(a,b)x(b,a) fixes the “lambda problem”. So, if we think that probabilities and transformation functions are both valid for describing unknowable mechanics, this is the way to unify the two ideas.
However, this means that x(a,b) must equal the complex conjugate of x(b,a). That is: the transformation functions map pairs of quality values to complex numbers.
Note that if you have x(a,b), you know p(b|a), but not the other way around. So, in some sense, the transformation functions are a more fundemental kind of knowledge about unknowable mechanics than probabilities. Since we are all used to thinking in probabilities, this is a rather disturbing fact: we’ve been thinking in the wrong space to get all the knowledge that is there.
So, we’ve identified a rather fundemental kind of knowledge we can have about the mechanics of machines we can’t know everything about. Since there are lots of qualities that have such mechanics, for instance:
-Polarizations that are at different angles
-Position and momentum
-Time and energy
-Value and ownership
-Current time and national product
this is pretty profound stuff. And we haven’t had to talk about sub-atomic particles at all to reveal it.
Next time we’ll talk about building toys with these ideas. Toys like Schrodinger’s Aibo.
