Playing with Quantum Toys: Julian Schwinger’s Measurement Algebra and the Material Culture of Quantum Mechanics Pedagogy at Harvard in the 1960s

Arthur Halprin

Quantum mechanics is hard. Can a toy make the subject any easier? Costas Papaliolios thought so. As a graduate student at Harvard in the early 1960s, he created these aluminum blocks marked with variations of Paul Dirac's bra-ket notation for describing quantum states. He made 4-inch blocks and 1-inch blocks. (The upper photos are enlargements of the 1-inch, or 2.54-centimeter, blocks.) A polarized filter fit inside each open-ended block. Shining a light through a block, or a series of blocks, yielded one of two results: The light passed through or it didn't. Papaliolios hoped his toys would help students grasp abstract concepts like linear operators and matrix representation. For most people, though, quantum mechanics remains as puzzling as ever.

Luisa Lovisetti, Marco Giliberti

Steven French

C. A. M. de Melo, B. M. Pimentel, Ramirez, J. A.

Here we deal in a pedagogical way with an approach to construct an algebraic structure for the Quantum Mechanical measurement processes from the concept of \emph{Measurement Symbol}. Such concept was conceived by Julian S. Schwinger and constitutes a fundamental piece in his variational formalism and its several applications.

Gregg S. Jaeger

Julian Schwinger provided to physics a mathematical reconstruction of quantum mechanics on the basis of the characteristics of sequences of measurements occurring at the atomic level of physical structure. The central component of this reconstruction is an algebra of symbols corresponding to quantum measurements, conceived of as discrete processes, which serve to relate experience to theory; collections of outcomes of identically circumscribed such measurements are attributed expectation values, which constitute the predictive content of the theory. The outcomes correspond to certain phase parameters appearing in the corresponding symbols, which are complex numbers, the algebra of which he finds by a process he refers to as 'induction'. Schwinger assumed these (individually unpredictable) phase parameters to take random, uniformly distributed definite values within a natural range. I have previously suggested that the 'principle of plenitude' may serve as a basis in principle for the occurrence of the definite measured values that are those members of the collections of measurement outcomes from which the corresponding observed statistics derive (Jaeger 2015Found. Phys.45, 806-819. (doi:10.1007/s10701-015-9893-6)). Here, I evaluate Schwinger's assumption in the context of recent critiques of the notion of randomness and explicitly relate the randomness of these phases with the principle of plenitude and, in this way, provide a fundamental grounding for the objective, physically irreducible probabilities, conceived of as graded possibilities, that are attributed to measurement outcomes by quantum mechanics.