# A Does Locality Entail Realism?

#### Dfpolis

In a paper much discussed on this forum (https://www.physicsforums.com/threads/does-the-bell-theorem-assume-reality.964219/),Roderich Tumulka, "The Assumptions of Bell’s Proof" (http://de.arxiv.org/abs/1501.04168) argues that locality entails two specific types of realism, which he calls R1 and R2. He defined these thus:
(R1) Every quantum observable (or at least (a · σ) ⊗ I and I ⊗ (b · σ) for every a and b) actually has a definite value even before any attempt to measure it; the measurement reveals that value.

(R2) The outcome of every experiment is pre-determined by some (“hidden”) variable λ.
Here I is the 2×2 unit matrix and σ the triple of Pauli matrices.

He gives the following counterfactual argument, which seems unsound on two counts that I will explain.
The EPR argument, for the experiment involving two spin-1/2 particles in the singlet state, can be put this way: Suppose that Alice and Bob always choose the z-direction, a = b = (0,0,1). Quantum mechanics then predicts that the outcomes are perfectly anti-correlated, A = −B with probability 1. Assume locality. Alice’s experiment takes place in a space-time region A and Bob’s in B at spacelike separation. There is a Lorentz frame in which A is finished before B begins; thus, in this frame, there is a time at which Alice’s experiment already has a definite outcome. She can therefore predict Bob’s outcome with certainty, although she cannot transmit this information to Bob before Bob carries out his experiment. Anyway, Bob’s outcome was already fixed on some spacelike hypersurface before his experiment. By locality, his outcome was not influenced by events in A, in particular not by whether Alice did any experiment at all. Thus, the state of affairs inside the past light cone of B, but before B itself, included a fact about the value Bz that Bob will obtain if he carries out a quantum measurement of I σz. In particular, Bz is a “hidden variable” in the sense that it cannot be read off from ψ. Since the argument works in the same way for any other direction b instead of z, there is a well defined value Bb for every unit vector b, such that if Bob chooses b0 then his outcome will be Bb0. Since the argument works in the same way for Alice, also her outcome just reveals the pre-determined value Aa for the particular direction a she chose.

We see how locality enters the argument, and how (R1), and thus (R2), come out. EPR’s reasoning is sometimes called a paradox, but the part of the reasoning that I just described is really not a paradox but an argument, showing that (L) implies (R1).
Why do I think this argument is invalid?

First, the conclusion, R1 is provably false. Consider the same experiment Tumulka has just described, but now have Bob and Alice align their spin detectors at right angles to each other. Then, no matter what spin they each measure, the vector sum of the measured spins cannot be zero. Still, we know that the initial state had zero angular momentum, so the sum of the measured spins is unequal to the initial angular momentum in apparent violation of conservation of angular momentum. We can easily save the conservation law by saying that the detectors contribute to the measured results, so that the measured values are the result of the interaction of the incident quantum with the detector. This violates R1, but not our common sense notions of realistic behavior. It just means that while quantum systems may have definite properties, actual measure numbers are the result of the details of the interaction of those properties with our detectors.

Since Tumulka's conclusion is false, there must be an error in the reasoning that gave rise to it. I see this error in the counterfactual nature of the argument. Somehow the world in which Alice does not perform her experiment must be differnt than the world in which she does. Can this be true without violating locality? Without Alice's experiment affecting the state of the world at Bob's location? I think it can, if Alice's observations can inform her of conditions at Bob's location. Then, if she performs her observation, she is informed, but if she does not, she is not informed.

How could an experiment at Alice's location inform us of conditions at Bob's location? Before I try to answer this, it is important to note that Tumulka has already conceded that it does. Once Alice has measured spin up, she knows that conditions at Bob's location are such that he will measure spin down. Further, it does not matter if she learns of the conditions at Bob's space-time location in a reference frame where she is informed before Bob does his measurement, or after. All that is important is that an observation at Alice's space-time location can inform us of conditions at Bob's.

Part of the answer is that one condition allowing this to happen spreads out from the origin of the experiment. If a spin-0 particle decays into two spin-1/2 particles, then the observation that informs Alice cannot take place before the spin-1/2 particle she observes arrives. So, the region of enablement does not expand faster than the speed of light. Yet, we know that the observed particle cannot carry sufficient information to explain the correlation between Alice's and Bob's observations. So, more is required.

All EPR type experiments involve entanglement via conservation laws. By Noether's theorem, this implicates correlative symmetries. So, might not the "something more" be constraints placed on Alice's and Bob's observations by symmetry? If we lived in a model world in which Bob's observations must mirror Alice's, then Alice would be able to predict Bob's results from hers without a hint of non-locality.

Of course, our world is more complex, but still, it embodies many dynamic symmmetries. One is the anti-symmetry of Fermion wave functions under interchange of coordinates. In Dirac's many-time formulation of relativistic quantum mechanics, this interchange involves the time as well as space coorinates, linking the multi-electron wave function, φ(x1, t1, x2, t2, ...), by a set of symmetry constraints, e.g.

φ(x1, t1, x2, t2, ...) = -φ(x2, t2, x1, t1, ...)

linking the wavefunction at all space time points. In principle, this transtemporal symmetry could inform Alice of detector conditions at Bob's location. At the very least, it makes the assumption of detector independence false -- giving hope for a manifest variable interpretation of quantum theory.

I await comments, pro or con.

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#### stevendaryl

Staff Emeritus
First, the conclusion, R1 is provably false. Consider the same experiment Tumulka has just described, but now have Bob and Alice align their spin detectors at right angles to each other. Then, no matter what spin they each measure, the vector sum of the measured spins cannot be zero. Still, we know that the initial state had zero angular momentum, so the sum of the measured spins is unequal to the initial angular momentum in apparent violation of conservation of angular momentum. We can easily save the conservation law by saying that the detectors contribute to the measured results, so that the measured values are the result of the interaction of the incident quantum with the detector. This violates R1, but not our common sense notions of realistic behavior. It just means that while quantum systems may have definite properties, actual measure numbers are the result of the details of the interaction of those properties with our detectors.
I don't see how that refutes $R_1$. You are assuming that the results are dependent on details of the measuring devices, but what is your basis for assuming that? I don't think it follows from conservation of angular momentum.

#### Dfpolis

I don't see how that refutes $R_1$. You are assuming that the results are dependent on details of the measuring devices, but what is your basis for assuming that? I don't think it follows from conservation of angular momentum.
Recall that R1 states: "Every quantum observable ... has a definite value even before any attempt to measure it; the measurement reveals that value." Surely, whatever the definite values of the two spins are, they must sum to zero. Thus, the measured values cannot both be the values prior to measurement. In order to maintain conservation, we must find a source for the difference in the initial and measured values of angular momentum. The only interactions that have occurred after the initiating event are with the detectors. They must be the source of the non-zero sum of measured values.

#### stevendaryl

Staff Emeritus
Recall that R1 states: "Every quantum observable ... has a definite value even before any attempt to measure it; the measurement reveals that value." Surely, whatever the definite values of the two spins are, they must sum to zero.
That doesn't follow. The only thing that follows from conservation of angular momentum is that if you measure both spins along the same axis, you have to get opposite results. If you measure them on different axes, there is no reason for the results to sum to zero.

Now, actually, $R_1$ is disproved by Bell's theorem. But not by the argument you're giving.

#### Dfpolis

That doesn't follow. The only thing that follows from conservation of angular momentum is that if you measure both spins along the same axis, you have to get opposite results. If you measure them on different axes, there is no reason for the results to sum to zero.

Now, actually, $R_1$ is disproved by Bell's theorem. But not by the argument you're giving.
Please give me an example prior values in which the measured values of an orthogonal set up are equal to the prior values, as required by R1. If you cannot do this, (1) we have no need of Bell's theorem to disprove R1, and (2) the argument given by Tumulka is unsound. If Tumulka's argument is unsound, then his understanding of the justification of Bell's theorem is unsound.

#### stevendaryl

Staff Emeritus
Please give me an example prior values in which the measured values of an orthogonal set up are equal to the prior values, as required by R1. If you cannot do this, (1) we have no need of Bell's theorem to disprove R1, and (2) the argument given by Tumulka is unsound. If Tumulka's argument is unsound, then his understanding of the justification of Bell's theorem is unsound.
R1 is false by Bell's Theorem. But YOU didn't prove it false, because your argument is invalid.

For example, suppose that one particle has spin $\frac{1}{2}$ in the x-direction, $\frac{1}{2}$ in the y-direction, and $\frac{1}{2}$ in the z-direction. The other particle has spin $-\frac{1}{2}$ in the x-direction, $-\frac{1}{2}$ in the y-direction, $-\frac{1}{2}$ in the z-direction. The total spin is zero, but if you only measure one particle's spin in the x-direction, and get $+\frac{1}{2}$, and measure the other particle's spin in the y-direction, and get $-\frac{1}{2}$, what does that tell you?

#### Dfpolis

R1 is false by Bell's Theorem. But YOU didn't prove it false, because your argument is invalid.

For example, suppose that one particle has spin $\frac{1}{2}$ in the x-direction, $\frac{1}{2}$ in the y-direction, and $\frac{1}{2}$ in the z-direction. The other particle has spin $-\frac{1}{2}$ in the x-direction, $-\frac{1}{2}$ in the y-direction, $-\frac{1}{2}$ in the z-direction. The total spin is zero, but if you only measure one particle's spin in the x-direction, and get $+\frac{1}{2}$, and measure the other particle's spin in the y-direction, and get $-\frac{1}{2}$, what does that tell you?
It means that you do not understand what a spin-1/2 particle is. The particle you describe would have a spin of 31/2/2.

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#### stevendaryl

Staff Emeritus
It means that you do not understand what a spin-1/2 particle is. The particle you describe would have a spin of 31/2/2.
I have the idea that you don't know what a "proof" is. You have to write down your assumptions, and then in a step-by-step fashion, you derive consequences from those assumptions and from previous conclusions, until finally the statement you're wanting to prove is derived.

So you want to assume that quantum spin works like classical vectors. Fine. Bell didn't assume that. What's your basis for assuming that?

In particular, you want to assume something along the lines of:

1. There is a vector $\overrightarrow{S}$ associated with each electron.
2. The magnitude of that vector is $\frac{\hbar}{2}$. (meaning that $\sqrt{(S_x)^2 + (S_y)^2 + (S_z)^2} = \frac{\hbar}{2}$
3. The measurement of a component of the spin relative to a direction $\hat{d}$ gives $\overrightarrow{S} \cdot \hat{d}$
4. Also, the measurement of a component of the spin relative to any direction always gives $\pm \frac{\hbar}{2}$

1-4 are certainly contradictory. So one of the assumptions is false. That means that electron spin doesn't behave like a classical vector. But that doesn't disprove the assumption that there are local hidden variables that give the values of spin measurements.

People knew before Bell that particle spin can't possibly act like a classical vector. That doesn't disprove the existence of local variables. It doesn't by itself disprove $R_1$.

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#### Dfpolis

I have the idea that you don't know what a "proof" is. ...

So you want to assume that quantum spin works like classical vectors. Fine. Bell didn't assume that. What's your basis for assuming that?
Thank you for your concern. Let me assure you, you are worrying needlessly.

Euclid prefaces his proofs with definitions, common notions and postulates. In physics, we do not usually define all of our terms, because they have accepted meanings that are understood by our audience. It is only when we reject or revise these definitions and common notions, or when we believe that our readers may not know them, that we explicate them. So, when one encounters terms such as "angular momentum" or "spin," one is justified in assuming that axial vectors are being discussed. Further, to apply conservation of angular momentum, it is necessary that each term summed for the initial and final states, including spin, have the same tensor character. If not, we are adding the proverbial apples and oranges.

Further, by Noether's theorem the symmetry underlying conservation of angular momentum is rotational invariance. Since rotation is described by an axial vector, it follows that angular momentum, as the canonically conjugate variable, must also be an axial vector.

Thus, when you say "that one particle has spin 1/2 in the x-direction, 1/2 in the y-direction, and 1/2 in the z-direction," you are writing a meaningless string unless "spin" has the meaning commonly accepted in physics and the directions are defined by the conventional unit vectors. Also, if the "spins" you are discussing are not axial vectors, it is unclear how to use them to determine the outcome spin observations along skew angles. My counterargument works with the detectors set to any non-parallel angle.

I am open to the possibility that quanta may have properties quite unlike those dealt with in classical physics, but that is irrelevant to the logic of Tumulka's argument.

In particular, you want to assume something along the lines of.:

....
3. The measurement of a component of the spin relative to a direction ^d gives →S⋅^d.
...
1-4 are certainly contradictory. So one of the assumptions is false. That means that electron spin doesn't behave like a classical vector. But that doesn't disprove the assumption that there are local hidden variables that give the values of spin measurements.
I am assuming, as is the custom in physics, that "spin" names the intrinsic angular momentum of a quantum, which is an axial vector. I also accept experimental results showing that electrons obey conservation of angular momentum if they have spin = ℏ/2, and that measurements of electron spin always yield a magnitude ℏ/2. I do not accept 3, which is the point at issue here.

Recall that what I am claiming is that Tumulka's argument for R1, is unsound, where

(R1) Every quantum observable (or at least (a · σ) ⊗ I and I ⊗ (b · σ) for every a and b) actually has a definite value even before any attempt to measure it; the measurement reveals that value.
In doing so, I am using the assumptions Tumulka makes in his argument. I conclude that the argument is unsound because these premises, applied to a different case, yield a contradictory conclusion.

Let me also note that your inference, "That means that electron spin doesn't behave like a classical [axial] vector," is unsound. It follows from my orthogonal case thought experiment, that, if "spin" enters the conservation equation, its measured value is a result of an interaction between the incident quantum and the detector. Surely there must be such an interaction for detection to occur and what we know of the incident quantum depends on this interaction. The same point was made by Heisenberg in his 1927 uncertainly paper.

#### stevendaryl

Staff Emeritus
I won't be responding further. I don't think that you're actually making a sensible argument.

#### Dfpolis

You are entitled to you beliefs. Thank you for your time.

#### PeterDonis

Mentor
You are entitled to you beliefs.
It's not a matter of "beliefs". Either you have presented a valid logical argument or you haven't. It seems like you haven't.

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