Bell’s Theorem & the EPR Paradox
Before attempting to explain what Bell’s theorem is said to prove, or not disprove, some initial context is required. Bell’s theorem is about the correlation of probabilities, which is often used as an argument as to which interpretation of quantum physics might be supported by the probability of actual measured outcomes. Unfortunately, most discussions quickly disappear into the mathematics of statistical probability with little further discussion of physical causality. As a consequence, it is often difficult to understand what the Bell theorem does, or does not support, within the framework of a quantum model. So, let us start with a general outline of how probability fits into this discussion.
Note: Probability predictions of Bells’ theorem can take the form of inequalities that are satisfied, or not, by correlations derived from a particular interpretation of quantum theory, but then apparently questioned, or violated, by correlations derived from measurements. These types of inequalities are known as Bell inequalities. This discussion will start from an initial assumption that Bell’s theorem shows that no ‘real and local’ theory can reproduce the probabilistic predictions of quantum mechanics under all circumstances. However, at this point, it is unclear whether this initial assessment is correct.
Naturally, most readers might want clarification of this somewhat ambiguous introduction of statistical probability and its ability to question different interpretations within quantum physics. However, if we accept that we are starting from a point of ignorance, this discussion is allowed to question, not prove, whether any of the assumptions associated with Bell’s theorem can really prove the validity of a given quantum interpretation. For it is possible that a violation of Bell inequalities may result from a lack of understanding of the actual causal nature underpinning quantum processes, which might be compounded by inaccuracies in measurement of experimental data.
Note: The idea that quantum theory is proven fact, beyond its probability predictions, might also be questioned, especially given multiple and conflicting interpretations. It might also be highlighted that the Bell inequalities primarily rests on mathematical logic without necessarily addressing the key issue of causality.
While the determination of the statistical probabilities underpinning Bell’s theorem can be a barrier to general understanding, it can also be compounded by the level of mathematical abstraction surrounding the quantum model itself. While the details of the quantum model are beyond the scope of this discussion, the following note may reflect some of the complexity being suggested.
Note: As a basic summary, the quantum model invokes many mathematical concepts, such as operators, matrices, commutative and non-commutative properties, normalisation, quantisation, group theory, quantum fields, wave function and collapse, relativistic and non-relativistic corrections along with numerous other assumptions. As a consequence, the quantum model has been built on increasing layers of assumptions, statistical probability and uncertainty about energy, time and momentum plus an ambiguity as to the physical reality of any subatomic ‘particle’. In this context, it is not necessarily unfair to say the quantum model has made numerous assumptions about variables and parameters that only have a tenuous connection to physical reality.
It is realised that this introduction is quickly deviating into issues
that are beyond the scope of Bell’s theorem of statistical probability.
However, there is an aspect of Bell’s theorem that requires an understanding
as to why quantum probability might be different from classical probability
that might be best characterised in terms of quantum entanglement.