Comparison Task – Bohm and Bell on Quantum Theory

The two uploaded PDF documents contain the following articles:

1. David Bohm: "A Suggested Interpretation of the Quantum Theory in Terms of 'Hidden' Variables I." (Physical Review, vol. 85, no. 2, 1952, pp. 166–179)
2. John S. Bell: "On the Einstein Podolsky Rosen Paradox." (Physics Physique Физика, vol. 1, no. 3, 1964, pp. 195–200)

Compare these two articles and list their opposing arguments, focusing particularly on their interpretations of quantum mechanics, the use of hidden variables, and the issue of non-locality. During the comparison, clearly indicate which arguments belong to which article.

Comparing Bell's and Bohm's Papers on Quantum Mechanics

These two papers represent pivotal contributions to the foundations of quantum mechanics, with Bohm proposing a hidden variable interpretation in 1952 and Bell responding in 1964 with his famous theorem showing limitations on such theories.

I. Core Arguments and Contributions

Bohm's Paper (1952)

• Main Thesis: Proposes a deterministic interpretation of quantum mechanics using hidden variables that could restore causality and completeness to quantum theory.
• Motivation: Challenges the Copenhagen interpretation's assertion that quantum mechanics is inherently probabilistic and complete.
• Approach: Reformulates Schrödinger's equation to describe a real physical field guiding particle motion, with determinate particle positions as hidden variables.

Bell's Paper (1964)

• Main Thesis: Proves mathematically that no local hidden variable theory can reproduce all quantum mechanical predictions.
• Motivation: Tests Einstein, Podolsky, and Rosen's (EPR) suggestion that quantum mechanics is incomplete and should be supplemented with additional variables.
• Approach: Derives an inequality (now known as "Bell's inequality") that must be satisfied by any local hidden variable theory but is violated by quantum mechanics.

II. Key Points of Comparison

1. On Hidden Variables

Bohm:

• Argues that hidden variables (specifically, precise particle positions) are both possible and desirable.
• Demonstrates an explicit hidden variable interpretation that reproduces quantum mechanical predictions.
• "In contrast to the usual interpretation, this alternative interpretation permits us to conceive of each individual system as being in a precisely definable state, whose changes with time are determined by definite laws."
• Views quantum-mechanical probabilities as practical necessities, not fundamental, similar to classical statistical mechanics.

Bell:

• Acknowledges that hidden variable theories (like Bohm's) are possible.
• "That particular interpretation has indeed a grossly non-local structure. This is characteristic, according to the result to be proved here, of any such theory which reproduces exactly the quantum mechanical predictions."
• Mathematically proves that local hidden variable theories cannot reproduce quantum correlations.
• Focuses on testing the locality assumption rather than arguing against hidden variables per se.

2. On Non-Locality

Bohm:

• "The 'quantum-mechanical' potential, U(x₁, x₂...xₙ)...acting on a particle is equivalent to that produced by a 'many-body' force, since the force between any two particles may depend significantly on the location of every other particle in the system."
• Accepts this non-locality as a feature of his interpretation.
• Does not emphasize this non-locality as problematic.

Bell:

• "It is the requirement of locality, or more precisely that the result of a measurement on one system be unaffected by operations on a distant system with which it has interacted in the past, that creates the essential difficulty."
• Proves that any hidden variable theory reproducing quantum mechanics must be non-local.
• "In a theory in which parameters are added to quantum mechanics to determine the results of individual measurements, without changing the statistical predictions, there must be a mechanism whereby the setting of one measuring device can influence the reading of another instrument, however remote."

3. On Completeness of Quantum Mechanics

Bohm:

• "The usual interpretation...involves an assumption that cannot be tested experimentally, viz., that the most complete possible specification of an individual system is in terms of a wave function."
• Maintains Einstein's view that "even at the quantum level, there must exist precisely definable elements or dynamical variables determining the actual behavior of each individual system."
• Considers completeness a key motivating factor for his alternative interpretation.

Bell:

• Takes no explicit position on completeness, focusing instead on locality.
• Tests the EPR argument that assumes both completeness and locality cannot simultaneously hold.
• Shows that locality must be abandoned regardless of the completeness question.

4. On Measurement Process

Bohm:

• Explains measurement as a causal, deterministic process with apparent randomness arising from our ignorance of precise initial conditions.
• Promises (in Paper II) to show that "our interpretation leads to precisely the same results for all experiments as are obtained with the usual interpretation."
• Speculates that modifications of his theory might eventually allow violations of the uncertainty principle.

Bell:

• Focuses on measurement correlations between entangled particles.
• Shows that regardless of the measurement details, local hidden variable theories cannot reproduce certain quantum mechanical correlations.
• "Experiments of the type proposed by Bohm and Aharonov, in which the settings are changed during the flight of the particles, are crucial."

III. Fundamental Differences in Approach

• Bohm: Constructive approach; philosophical orientation seeking to restore determinism and causality; forward-looking.
• Bell: Analytical approach; rigorous, inequality-based framework; result-focused.

IV. Historical Impact

The interplay between these papers fundamentally shaped quantum foundations research. Bohm showed that hidden variable theories were possible, challenging the Copenhagen orthodoxy. Bell then proved that such theories must be non-local, leading to experimental tests that have consistently supported quantum mechanics and non-locality. Rather than being purely opposing, Bell's work can be seen as building upon Bohm's, identifying the unavoidable non-locality in Bohm's theory and proving it must be a feature of any hidden variable theory that reproduces quantum predictions.