Questions about Empirical Equivalence in STR and QT

No, because the equations instantiate a certain mathematical structure which (at least in some sense) includes the Minkowski metric, whether that structure is explicitly adverted to or not. There are different ways of writing the equations, but those ways refer to the same measurable quantities and share the same structure, and so really are the same equations. (And indeed, the that you choose to write the equations need not be dictated by your interpretation of the theory.)

Oh, I think I get it. It’s been a while, but I recall in high school math doing stuff with fractions or in algebra in solving problems to rework the problems in a way that made them easier to work with. Though not exactly the same, in essence the equations were the same, just an easier to work with version so to speak. Is that sort of the idea?

… sort of, yeah. Another example would be alternative formulations of Maxwell’s equations. These two sets of expressions:

image

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represent exactly the same information, but only the second set is explicitly cast in four-dimensional spacetime, while the first set handles space and time separately.

When you say exactly the same information I’m assuming you mean something akin to:
3x+2=14 is the same information as x=4. So the bottom spacetime equation is the same information as the top space + time equation just in a more condensed, concise, or reduced form. And there’s no need to go through the steps of the top equation when the same can be accomplished in one step with the bottom equation similar to not having to go through the steps 3x+2−2=14−2 3x=12 (3x)/3=12/3 to get from 3x+2=14 to x=4 if we already know in advance that x=4. Is that sort of somewhere in the ball park?

Do you know about matrices? It’s more along the lines of the way a whole system of linear equations can be represented as a single matrix equation.

No I don’t. But I’ll try and familiarize myself with it and get back to you.

So in the case of physics is the matrix being used in a sense to map an object in respect to its location in space and the numbers represent relative coordinates of the object within the particular space that the matrix represents? If not, what do the numbers within the matrix represent?

Matrices are way too general for there to be a single answer to that question.

It’s essential to know some linear algebra in order to understand quantum mechanics, so I recommend you look into this subject more. There’s a YouTube channel called 3Blue1Brown that has a really excellent playlist giving a conceptual introduction to linear algebra, vectors, and matrices - you should check it out!

OK. Already watched on before you posted. Will work on watching more and get back to you.

OK. I’ve watched several of the videos and I think I get the gist of what’s going on. So would I be correct to say that when you say “the same information” you’re referring to something like what happens in a linear transformation where the “transformed” space retains the same information as the “basis” space in regards to the relationship between the vector coordinates and the x y axis basis coordinates of i and j?

No, that’s not it. Refer back to the two formulations of Maxwell’s equations that I posted above. One is written using 3-dimensional vectors and handles space and time separately; the other is written using 4-dimensional quantities and handles space and time in a unified way. The two sets of equations represent the same information; very similarly to how a system of linear equations such as:

4x+3y+z = 1
-x+2y-z = 0
x+3y+3z = 4

can be written as a single matrix equation:

Ax = b

(where x is a vector representing all three of the variables x, y, and z at once, and A and b are appropriately chosen to represent the coefficients of the variables on the left-hand side of the equation, and the constants on the right-hand side of the equation, respectively).

That was the comparison I was making. That’s a slight tangent from the point I was trying to make, which is that one of the ways of writing Maxwell’s equations that I posted above invokes space-time structure explicitly (by writing it in terms of 4-dimensional quantities), while the other way does not invoke that structure explicitly. However, it does have that structure implicitly - which we can see by the fact that it contains the same information and therefore in a sense instantiates the same abstract mathematical structure as something that has that structure explicitly.

And in the same way as that, Einstein’s first formulation of SR, which does not explicitly advert to the geometry of space-time, nevertheless implicitly contains the same mathematical structure as later formulations which bring out the geometry of Minkowski space-time explicitly.

ax+by+cz=x
dx+ey+fz=y
gx+hy+iz=z

So would A represent the ijk coordinates, x the xyz inputs, and b the xyz outputs?

If so, I think I’m getting it. So in essence the upper equations are in a sense algebraically condensed to get the lower equation that represents the upper equations, so to speak.

Again, no, not quite right. For one, you’re trying to apply a specific interpretation to the example I gave which was not intended to have any interpretation - it’s just a system of linear equations. For two, in the interpretation you’re trying to give it (as far as I can tell) what you should say is something like:

A represents the transformation between the coordinate systems, x is the point in the original coordinates, and b is the point in new coordinates.

And for three, what you wrote above is actually a system of equations like Ax=x, not Ax=b. That’s a slightly different kind of equation.


With all of that said though,

This is correct.

OK. Thanks for clarifying that. So would it be correct to say that the 3 versions of STR–Lorentz’s, Einstein’s original formulation with space + time, and the later formulations with Minskowskian spacetime–have their own separate and distinct postulates, but the differences from the progression of the first equations up till today are simply different ways of expressing what are foundationally the same equations?

Edit: I forgot, you also seem to argue that the postulates are only different in an ontological sense, not in terms of empirical quantities. Right?

More or less, if not exactly how I would put it. (For how I would put it, see my first reply to the thread!)

Sometimes I like to sit back and watch high-level discussions go SWOOSH above my head. :wink:

Ok. I think I see where you’re coming from on SRT. And though I might still have questions about what I think you’re saying in regards to the relationship between mechanism and causality, I think generally it makes sense to me.

Just so I’m clear, with QT you are saying that among various competing interpretations there are those that have the Schrodinger equation in common. The Born rule seems to be unique to orthodox QT.

Apart from that there are various competing interpretations that contain other equations, used with or without Schrodinger, that on their own are unique, and not just different ways of expressing what, in the end, are foundationally the same equation, correct?

The Born rule isn’t unique to orthodox QM, but many interpretations try to do without it (at least on a fundamental level) or derive it from other postulates.

Correct. There’s nothing in the many-worlds interpretation, for example, that corresponds to the guidance equation from pilot-wave theory.

OK. So that brings me to the question about empirical equivalence. If there are empirically equivalent theories in QT and STR with different postulates, and in QT with both different postulates and equations, how does that comport with the idea that seems to be prevalent that the conceptual framework of theories in those two fields represents a possibility of what reality is like, and that the most successful of those theories, assuming that the most successful theory is based on the confirmation and accuracy of its predictions, would correspond to the most probable of what actual reality is like? In the case of empirical equivalence all the theories are equally successful, are they not?

If I’m understanding your question, you are basically alluding to the problem of underdetermination for scientific realism. (See also on Wikipedia.) If there are two or more empirically equivalent theories, the evidence alone can’t determine one way or the other which of those theories is more likely true. In that case, I would say we need to turn to philosophical evaluation of those theories to judge them on further grounds (e.g. coherence, parsimony, explanatory power).

Taking a more moderate stance on scientific realism (e.g. epistemic structural realism) reduces the force of this problem as well.