Structural Causality

Using quantum entanglement as a simple model of correlations between two systems, one can show that a new causal category must be incorporated into any complete description of physical reality. Such “structural causality” is related to the structural properties of the product space underlying a functional description of a multipart entity. Aristotelian causal categories and the restricted derivative notion of causality prevalent in modern science necessarily involve an exchange of matter and energy between causally connected events if change is to occur. Structural causality is fundamentally different in that no matter-energy connectivity is required for events to be causally linked providing that the underlying entities are suitably entangled. Aristotelian Causal Categories Aristotle recognized four causal categories: the material and the formal spring from, simply, matter and form. To these two obvious causes, Aristotle added efficient cause and final cause, but recognized that all four causes may coincide in particular instances. To know is to know by means of causes; causes are answers to “why?” questions. In a relational description of an organism, the organism is closed to efficient cause [Rosen, 1991] and the material cause can be identified with the final cause; that is, the answer to the question “why an organism?” is based in the physical properties of matter and energy and needs no other explanation. The ensuing questions considered relevant to science are descriptive and historical only: how matter and energy interact to give rise to complex forms and what particular historical path led from undifferentiated energy (Schopenhauer’s ñ¢••e) to the manifest and overwhelming complexity of the observer and observed (Schopenhauer’s “subject” and “object”). 18 and 19 Century Causality Humean empiricism, so prevalent in modern science, makes do with a single causal category, one that Hume identified with contiguity in space and succession in time, a property that we would call correlation. Hume’s skepticism could find no demonstrable way to infer the existence of one object from that of another, leading to the conclusion that, beyond correlation, causality is a construction of the mind, a simple association of ideas. The inability of reductionist science to deal with living organisms is a direct result of this impoverished causal scheme. Kant went beyond Hume to argue that we do not acquire our ideas of space and time by reflecting on the empirically given since the very description of such empirical situations presuppose familiarity with both space and time. Thus, the notions of space and time are a priori intuitions (a Kantian term), that are built in, as it were, by the evolutionary process, as one might say today. Schopenhauer, following Kant, made these arguments stronger and more forceful. Schopenhauer’s starting point was the simple observation taken from Kant [Schopenhauer, 1969] that “time and space, however, each by itself, can be represented in intuition even without matter; but matter cannot be so represented without time and space.” Schopenhauer evidently gives primacy to the material cause when he states that “the subjective correlate of matter or of causality, for the two are one and the same, is the understanding, and it is nothing more that this.” [Schopenhauer, p. 11]. He goes on to state that “the first, simplest, everpresent manifestation of understanding is perception of the actual world.” Thus, causality, like time and space are a priori categories built in to human perception. This comprehensive view of causality subsumes Hume’s skeptic approach while not ruling out intuitive and speculative insights. 20th Century Causality In the 20th century, with the advent of quantum mechanics and relativity, the view of causal connections changed to a relationship between entities located in space-time. Entities may be taken to include matter and radiation, both forms of energy and manifestations of Schopenhauer’s ñ¢••a. The last vestiges of Aristotle’s causal categories was removed from science by black-listing the question: ‘Why?’ allowing scientists formulate answers only to ‘How?’ An example is Feynman’s famous remark on the understanding of quantum mechanics where he posed the rhetorical question, “But how can it be like that?” and answered, “Nobody knows how it can be like that.” Both the question and its answer are clear uses of “why” presented in the guise of “how.” For example, in plain English, one would simply say, “Nobody knows why [quantum entities behave as they do].” Rosen [1987] recognized that the Aristotelian causal categories must be brought back into science if physics and biology were ever to deal consistently with organisms. The ability to ask ‘why’ must be reintroduced into scientific investigations. To be sure, the often contentious and pointless questions and arguments found in the past under the general label “teleology” must still be avoided. Such ‘why’ questions are still best left for poets to ponder, but Rosen [1987, 1991, 1999] pointed the way toward a rational and meaningful way to ask why within the framework of science. Einsteinian Causality Einsteinian causality restricts the Humean notion to events that can be connected by a ray of light (events lying within the so-called light cone). The prevalent view in modern physics harkens back to the naïve view of Humean correlation, but a correlation restricted to the light cone. It is tacitly assumed but seldom directly discussed, while maintaining consistency with the concept of the forward light cone, that two events must also actually be connected by matter or energy transfer to be considered causal; a direction is also imposed by the time asymmetry between the forward light cone (the future) and the backward one (the past). This notion of physical causality goes beyond the Hume’s idea and permits an extension of the causal beyond mere correlation. Einstein, in his famous 1935 paper [Einstein, Poldosky, and Rosen, 1935], opened the door to a new causal category based on quantum entanglement. Quantum entanglement is simply described on the tensor product of two or more Hilbert spaces and is presented in some detail in a following section. Causality in Physics Since the classical view of causality presupposes the forbidden question: “Why?”, contemporary science limits use of the word and concept erstwhile found in “causality” to an impoverished version of Hume’s empirical view. Causal connections are viewed in much the same way as James Burke’s Connections: amusing and educational but hardly serious science. The notion of causes are replaced by phase-space trajectories and their temporal evolution; as a system evolves in time, its phase-space point moves along a trajectory. The ‘cause’ of the present state is nothing more than its preceding state and the equations of motion. Such a view is computationally fruitful and has lead to amazing advances in science and engineering over the past 300 years. However, this view of causality, while fruitful, presupposes that the system under consideration is described by a Newtonian view to sufficient accuracy to explain (understand?) the relevant behaviors. If one loses sight of this underlying assumption and blindly applies the Newtonian state-space “paradigm” to an arbitrary natural system, confusion and contradiction can result. An example is to view the energy-transfer mechanisms of a cell under the umbrella of equilibrium thermodynamics (the classic example of a state-space, system-trajectory model); a result is the conclusion that muscle efficiencies are greater than 100%, clearly an error if the Second Law (of thermodynamics) has any meaning. Quite often, understanding is relegated to the realm of “physical intuition” and replaced with view that the differential equation defining the trajectory in phase space is somehow the primary ontological object. World-class scientists have both an extraordinary physical intuition or native understanding of how things work and a highly developed skill in state-space concepts and formalisms; they are able to avoid most of these conceptual traps. While understanding causal connections, they are able to work within the framework that restricts causality to correlations. Structural Causality To formulate the ideas underlying structural causality and to support the assertion that it represents a separate and novel causal category, a few of the basic concepts of quantum mechanics and some of the mathematics of Hilbert spaces must be presented. The formalism presented below is an excerpt from a more extensive development [Dress, 1999], and contains just enough detail for the reader to grasp the essentials of entanglement from a mathematical and physical perspective. The “essence” of quantum mechanics, that is where it differs from classical theories, lies in the behavior of amplitudes, how amplitudes are combined, and how to derive testable predictions, including statements of probability, from these amplitudes. Perhaps the most dramatic departure from any classical theory is to be found in the predicted and observed correlations between quantum entities such as electrons, photons, or atoms that are “entangled.” In fact, quantum interference and uncontrollable disturbances described and predicted by the Uncertainty Principle pale beside the behavior of two entangled photons or electrons. Why? Simply because interference phenomena and the Uncertainty Principle have well-understood analogues in classical physics whereas the behavior of two entangled quantum particles has no counterpart—unless one is willing to admit magic or faster-than-light actions-at-a-distance. Entangled Dice—A model of entangled entities Steps sufficient to obtain entangled states involve simple linear combinations of eigenfunctions of operators on finite-dimensional Hilbert spaces. The interpretation of the square of the absolute value of the complex coefficients in such a linear expansion as a probability measure leads directly to a joint probability table of observing, or obtaining in an experiment, particular states or eigenvalues. The example for this development will be “magic” dice that have from 1 to k sides with each side distinctively marked. This heuristic allows one to maintain a concrete picture of discrete objects with discrete states. Each die, or coin when k = 2, is represented on a k-dimensional Hilbert space. The eigenvalues and functions belong to an operator on the product Hilbert space (one Hilbert space for each die) and may be identified with observing the ‘up’ faces of dice in a tossing game. The behavior of entangled dice as well as quantum particles is simply and easily explained while their existence may be inferred from experimental observations on a dice game or experiments with photons or electrons. Quantum mechanics on Hilbert space Define a Hilbert space H as a complete inner-product space over the complex numbers. The completeness property, which assures convergence in H of sequences of elements of H, is required for dealing with infinite-dimensional spaces necessary for the complete development of quantum mechanics. Since the dice of interest have a finite number of sides, completeness will not be used in the following development and the proofs will be limited to the finite-dimensional case. The term “vector” will be used interchangeably with “function” when referring to elements of H. The complex-number field is essential to describe interference phenomena; as the examples presented below do not require interference, the phases (arguments) of the complex numbers are generally ignored. Any vector in H may be expanded as a linear combination, with complex coefficients, of a suitable basis set of H. Dirac notation is a convenient way to represent vectors and operators in a Hilbert space, for example, »a\ represents a vector with label a (usually an eigenvalue of some operator) while Xa» represents the corresponding covector. The inner product of two vectors is represented by Xa » b\ = Xb » a\* , where * is complex conjugation. If {»j\} is a basis of H, then any vector in H may be represented by a linear expansion on that basis as » y\ = ⁄ j c j » j\, where the 8cj< are complex numbers. If »y\ is normalized so that Xy » y\ = 1, then ⁄ j c j c j 2 = 1. Define (Bayesian) probability as a mapping from a denumerable set of statements X to the interval [0,1] such that p(x) ≥ 0, p(«) = 0, and p(X) = 1. The symbol, p(x), read “the probability that statement x is true” or , simply, “the probability of x,” is always assumed to be conditioned on some set of circumstances such as an experimental arrangement. This conditioning or contextuality may be explicitly given by the notation x|y which is read “x given that the statement y is true.” The mapping p obeys the product and sum rules [Cox, 1946], which state that p(xy) = p(x) p(y|x) and p(x+y) = p(x) + p(y) p(xy), where x,y œ X, “x+y” is the disjunction of the two statements and “xy” or “x,y” is their conjunction. Note that both conjunction and disjunction are associative and commutative, and that conjunction distributes over disjunction as in ordinary arithmetic. A probability distribution {pj : pj œ [0,1]} is a particular assignment to, or a mapping from, the elements xj of a denumerable set of statements, to the interval [0,1] such that pj = pHx jL. In quantum mechanics, the physical state of a quantum system is represented by a unique direction in Hilbert space; all proportional vectors represent the same state. The concept of a state comes from Newtonian physics and is taken to be a primitive notion. The postulate tells us that the vector a|y\, where a ≠ 0, and |y\ represent the same state. Any physical observable of a quantum system is represented by a linear, self-adjoint operator on the corresponding Hilbert space, and the square of the absolute value of an amplitude of a normalized vector in a Hilbert space is the probability of obtaining the eigenvalue belonging to that eigenvector in a measurement of the corresponding operator. These postulates of quantum mechanics and the definition of a probability distribution allow one to associate a probability distribution on the discrete set of basis functions or eigenvalues. Suppose the probability of obtaining the eigenvalue lm belonging to the eigenfunction fm in a measurement of the observable A on the state |y\ is pHlmL, then this probability distribution is given by (1) pHlkL = » Xfm » y\ »2 . To take a specific example, consider a die with k sides. The physical-state postulate lets us represent the state of this possibly actual object by a vector in a suitable Hilbert space. If the observable of interest is the number of spots appearing in a toss of the die, the space has k dimensions and we assume that the number of spots that can show in a given toss is an element of the set Zk={1,..., k}. Of course, spots as distinguishing features are arbitrary, so one could have colors or engraved marks or any other symbol set to distinguish between the sides; the assumption is that we can distinguish between each of the k sides in any toss. Tensor products of Hilbert spaces The single observable on a single Hilbert space adequately represents a game situation where one die is tossed: the states of a single die belong to a simple Hilbert space as shown in the previous section. To describe a game with more than one die, a larger space is needed. The states of a set of dice in a single toss can also be considered as states in a new Hilbert space. The new space is a direct product of the individual, simpler spaces, one associated with each die. For n dice of k sides each, the product space has k n dimensions. Define the direct or Cartesian product of two vector spaces, H1 and H2 as the set of all ordered pairs »a\»b\ where the first factor belongs to the first space and the second to the second. Such pairs are to represent vectors in the Hilbert space denoted by H1 ≈H2. A state consisting of two dice is then written as (2) » y\ = ‚ ci, j ... i] ... j], and we can now ask questions of the form, “What is the probability, in a single toss of a pair of dice, of observing eigenvalue m for one die and eigenvalue n for the other?” The answers to such questions are joint probabilities for the joint occurrences of the two states in the same experiment or toss. The joint probability for obtaining eigenvalue m for the state on H1 and the eigenvalue n for the state on H1 is given by (3) pHm, nL = » Xm, n » y\ »2 = ... ‚ ci, j Xm » i\ Xn » j\ ...2 = » cm,n »2 It may appear that multiple, k-sided dice are quite boring if they are fair (fairness implies uniform marginal probabilities). Note that the implication chain ‘uniform conditionals’Ø‘uniform joint’Ø‘uniform marginals’Ø‘dice are fair’ is valid as is the converse starting with ‘unfair dice.’ However, it is easy to show that the assumption of fair dice does not imply uniform conditionals with the single counterexample of the joint probability for two quasi coins given by (4) Pe1,e2 = 1 ÅÅÅÅ 2 i k jj sin2 q cos2 q cos2 q sin2 q y