12-22-2015, 05:14 PM

First of all, there is the basic idea of positivism: Science is about what we can observe. Thus, hidden variables, once we cannot observe them, are not part of science. In more radical versions, they simply do not exist.

But positivism is not only a wrong philosophy of science, it is even no longer the on preferred by the mainstream of modern physics. (Unfortunately, it has yet a strong influence, and a negative one - but this is another question.)

The mainstream philosophy of science is Popper's fallibilism. It is based on Popper's criterion of demarcation of empirical science, that a scientific theory has to be falsifiable: An empirical theory has to make empirical predictions, predictions which can be tested, and which can, in principle, appear false. If this happens, the theory will be falsified. If not, it is not proven - it is only corroborated - but it appeared to be a useful tool, a tool which allowed to make correct predictions. Which does not prove that this will remain in future too. The next experiment can, possibly, falsify the theory. And a theory which can be falsified tomorrow is, of course, not a proven theory.

It is also worth to be noted that this criterion is not only an absolute one - if the theory does not make falsifiable predictions, it is not an empirical theory - but also a relative one. All the falsifiable empirical predictions made by a theory define its empirical content. And the empirical content of one theory may be greater than that of another theory: If all predictions of theory A are also predictions of theory B, but theory B has at least one prediction which is not also a prediction of theory A, then B has a greater empirical content, and is, therefore, preferable according to Popper's criterion of empirical content.

What does this mean for Hidden Variable Theories? First of all, if one looks at them roughly, they appear now on equal foot. The Hidden Variable Theory uses the same equations, thus, usually makes the same predictions as the variant without hidden variables. So, all empirical predictions of above variants seem to be the same. This means no difference in the empirical content. Popper's criterion remains silent. That means, above theories are on equal foot. No argument against the Hidden Variable Theory.

Additional Predictions

But if one looks more carefully, then one finds that one can, sometimes, find some additional arguments. The point is that even if the equations of the different interpretations of one and the same theory are the same, there may be nonetheless some minor differences in the predictions, differences which depend on the Hidden Variables.

The point is that the Hidden Variables often give some additional structure. This structure may exist - or not. For some solutions of the equations, it may appear impossible to introduce this additional structure. But that means that this solution, if observed, would falsify the Hidden Variable Theory, without falsifying the theory without them. And that would give the Hidden Variable Theory additional empirical power.

An example is the ether interpretation of the Einstein equations of GR. This interpretation requires the existence of absolute space, \(\mathbb{R}^3\) and absolute time \(\mathbb{R}\). So, the ether theory already fixes the topology of the solution, it has to be trivial, \(\mathbb{R}^3\times\mathbb{R}\). All solutions with other topology - like wormholes - are simply not allowed in the ether interpretation. If we observe a wormhole, the ether interpretation would be falsified, the spacetime interpretation not, so that the ether interpretation gives additional predictive power.

Even more, in the ether interpretation, absolute time has to be a time-like coordinate everywhere, globally. This excludes some more solutions, like Gödel's rotating universe, which contains closed causal loops. Details can be discussed here.

Another, more subtle difference can be found between the Lorentz ether and the Minkowski spacetime interpretation of SR, in relation to the violation of Bell's inequality. Here one has to think about the status of faster-than-light causal influences. In the spacetime interpretation, they are forbidden, completely. Because whenever we have a faster than light causal influence, one can choose another system of inertial coordinates so that this causal influence goes into the past. Causal influences into the past do not make sense, and every inertial frame is as good as any other, thus, such causal influences have to be forbidden.

But does this argument hold in the Lorentz ether? No. In the Lorentz ether, there is only one frame where we have some connection between time and causality - the rest frame of the ether, where the time coordinate is absolute time. Causal influences backward in time, for this absolute time, have, of course, to be forbidden. But what about all the other time coordinates? Following the Lorentz ether interpretation, they have no fundamental importance at all, they are simply strange coordinates. So, if a causal connection is going backward in the "time" of such an irrelevant time coordinate, this does not matter at all. What matters is only if it is an influence into the future in true time.

Thus, the Lorentz ether does not forbid hidden causal influences faster than light. But, even if this is only relevant for hidden causal influences, it appears relevant for observable predictions - it makes a difference in the proof of Bell's inequality. The situation is reverted here: It is the spacetime interpretation which allows to prove Bell's inequality, but the Lorentz interpretation seems not sufficient for this. So, the spacetime interpretation seems to have here the higher empirical content.

Unfortunately for the spacetime interpretation, Bell's inequality is violated in quantum theory, and there seems to be sufficient empirical evidence that it is violated in reality too. This question can be discussed here.

But positivism is not only a wrong philosophy of science, it is even no longer the on preferred by the mainstream of modern physics. (Unfortunately, it has yet a strong influence, and a negative one - but this is another question.)

The mainstream philosophy of science is Popper's fallibilism. It is based on Popper's criterion of demarcation of empirical science, that a scientific theory has to be falsifiable: An empirical theory has to make empirical predictions, predictions which can be tested, and which can, in principle, appear false. If this happens, the theory will be falsified. If not, it is not proven - it is only corroborated - but it appeared to be a useful tool, a tool which allowed to make correct predictions. Which does not prove that this will remain in future too. The next experiment can, possibly, falsify the theory. And a theory which can be falsified tomorrow is, of course, not a proven theory.

It is also worth to be noted that this criterion is not only an absolute one - if the theory does not make falsifiable predictions, it is not an empirical theory - but also a relative one. All the falsifiable empirical predictions made by a theory define its empirical content. And the empirical content of one theory may be greater than that of another theory: If all predictions of theory A are also predictions of theory B, but theory B has at least one prediction which is not also a prediction of theory A, then B has a greater empirical content, and is, therefore, preferable according to Popper's criterion of empirical content.

What does this mean for Hidden Variable Theories? First of all, if one looks at them roughly, they appear now on equal foot. The Hidden Variable Theory uses the same equations, thus, usually makes the same predictions as the variant without hidden variables. So, all empirical predictions of above variants seem to be the same. This means no difference in the empirical content. Popper's criterion remains silent. That means, above theories are on equal foot. No argument against the Hidden Variable Theory.

Additional Predictions

But if one looks more carefully, then one finds that one can, sometimes, find some additional arguments. The point is that even if the equations of the different interpretations of one and the same theory are the same, there may be nonetheless some minor differences in the predictions, differences which depend on the Hidden Variables.

The point is that the Hidden Variables often give some additional structure. This structure may exist - or not. For some solutions of the equations, it may appear impossible to introduce this additional structure. But that means that this solution, if observed, would falsify the Hidden Variable Theory, without falsifying the theory without them. And that would give the Hidden Variable Theory additional empirical power.

An example is the ether interpretation of the Einstein equations of GR. This interpretation requires the existence of absolute space, \(\mathbb{R}^3\) and absolute time \(\mathbb{R}\). So, the ether theory already fixes the topology of the solution, it has to be trivial, \(\mathbb{R}^3\times\mathbb{R}\). All solutions with other topology - like wormholes - are simply not allowed in the ether interpretation. If we observe a wormhole, the ether interpretation would be falsified, the spacetime interpretation not, so that the ether interpretation gives additional predictive power.

Even more, in the ether interpretation, absolute time has to be a time-like coordinate everywhere, globally. This excludes some more solutions, like Gödel's rotating universe, which contains closed causal loops. Details can be discussed here.

Another, more subtle difference can be found between the Lorentz ether and the Minkowski spacetime interpretation of SR, in relation to the violation of Bell's inequality. Here one has to think about the status of faster-than-light causal influences. In the spacetime interpretation, they are forbidden, completely. Because whenever we have a faster than light causal influence, one can choose another system of inertial coordinates so that this causal influence goes into the past. Causal influences into the past do not make sense, and every inertial frame is as good as any other, thus, such causal influences have to be forbidden.

But does this argument hold in the Lorentz ether? No. In the Lorentz ether, there is only one frame where we have some connection between time and causality - the rest frame of the ether, where the time coordinate is absolute time. Causal influences backward in time, for this absolute time, have, of course, to be forbidden. But what about all the other time coordinates? Following the Lorentz ether interpretation, they have no fundamental importance at all, they are simply strange coordinates. So, if a causal connection is going backward in the "time" of such an irrelevant time coordinate, this does not matter at all. What matters is only if it is an influence into the future in true time.

Thus, the Lorentz ether does not forbid hidden causal influences faster than light. But, even if this is only relevant for hidden causal influences, it appears relevant for observable predictions - it makes a difference in the proof of Bell's inequality. The situation is reverted here: It is the spacetime interpretation which allows to prove Bell's inequality, but the Lorentz interpretation seems not sufficient for this. So, the spacetime interpretation seems to have here the higher empirical content.

Unfortunately for the spacetime interpretation, Bell's inequality is violated in quantum theory, and there seems to be sufficient empirical evidence that it is violated in reality too. This question can be discussed here.