On the “repeated addition” problem: ripping the heart out of mathematics and children

It might be useful to tell students that multiplication can be seen as repeated addition under certain circumstances. But docking a student half credit because he/she used commutativity when that was allegedly not permitted–because not specifically taught yet–is fallacious. You can see why if you simply look at a picture of repeated addition for the stated problem (5X3=…), say using stars:

15 stars

To get the so-called ‘correct solution’ you repeat the addition of the rows, and to get the so-called ‘wrong solution’ you repeat the addition of the columns. Docking half credit amounted to telling the student they could not add the columns. Unfortunately many children in this position simply assume, in total bewilderment, that math is some inscrutable strange mystical language that they will never hope to figure out.

The student was absolutely correct in his/her answer, whether or not anyone taught that student the concept of commutativity. Prohibiting the use of commutativity in this solution amounts to discounting and even prohibiting the picturing of multiplication, whose far more comprehensive and direct meaning is that of area. The teaching of multiplication as repeated addition dependent on the order in which numerals are written down on paper shrinks–and distorts–the concept of multiplication down to a shadow of what it really is. All this confusion could be avoided if students are encouraged to picture math concepts instead of thinking of them as a matter of reading symbols from left to right and following a ‘strategy’ (one that doesn’t work when dealing with concepts like the area of a circle which cannot be solved by repeated addition anyway).

Furthermore, clearly the intent of the problem was for the student to literally write down the meaning of “5X3,” which in the context of repeated addition was assumed to be “the number 3 taken 5 times”. But this is just an arbitrary linguistic convention. The expression “5X3” could also linguistically mean “the number 5 taken 3 times”. Indeed many of us, when ordering an item, list the item on the left and then put a numeral in a box to the right saying how many of that item we want! So in that case the problem is clearly saying ‘we want 3 fives’. This is perfectly natural. So it is not at all clear that the intended grading of this problem is even assessing use of commutativity, as claimed in many defenses of the grading. The student could simply have a different linguistic interpretation of the expression. And it’s completely appropriate. In my opinion, this student and all other students marked wrong on this problem are owed an apology. And a different way of teaching multiplication that shows what it is conceptually rather than reducing it to linguistically arbitrary ‘strategies’ that don’t necessarily work for all kind of numbers (e.g., irrational or transcendental numbers).

And if you’re worried about non-commutativity, I’m sure that students who are not totally discouraged by repeated addition conventions will be able to deal with it when they get to quantum mechanics and/or non-Abelian group theory. There are pictures for that too. Here’s one for non-commuting operators in quantum theory. There’s a big difference between (1) first opening the window and then sticking your head out; and (2) first sticking your head out and then opening the window.

Where did this ‘Wrong’ idea of quantum theory implying consciousness come from? Quantum physicists.

There has been much angst in the cybersphere recently about purported hijackings of solid, rational physical theory in service of ‘unprincipled New Age fantasies’ about ‘Consciousness’ being implied by quantum theory. The purpose of this post is to set the record straight about where these allegedly  ‘Crazy’, ‘Wrong’ ideas came from: distinguished pioneering quantum physicists. In fact, this is all ancient history for students of foundations of physics. It can be found in the comprehensive historical record of the pioneering discussions of the implications of quantum theory, Quantum Theory and Measurement (a collection of essays edited by Wojciech Zurek and Nobel Laureate John A. Wheeler), which I’ll abbreviate here as QTM.

Before I get into that, however, a caveat: my proposed interpretation of quantum theory, the ‘Possibilist Transactional Interpretation’ (PTI), (account for the general reader here) provides an observer-independent account of quantum measurement. PTI accounts for the measurement process without any necessary reference to an ‘outside conscious observer.‘ (This was of course also true of the original Transactional Interpretation (TI) of John Cramer; my work is just an extension of TI.) The issue of how to account for conscious experience then returns to the realm of metaphysics (and philosophy of mind and  psychology) where it belongs. In saying that, however, I do not disparage metaphysics; I recognize it as a legitimate realm of inquiry. And quantum theory can be interpreted as having some bearing on such questions, even though consciousness is not an absolute requirement for describing the process of measurement itself, as shown by the TI formulation which takes absorption into account.

Now let’s look at the history of the development of quantum mechanics, which was thoroughly saturated with discussions of consciousness and the mind. First,  celebrated mathematical genius and quantum theory pioneer John von Neumann stated in 1955 that “N. Bohr, Naturwiss. 17 (1929)…was the first to point out that the dual description…necessitated by the formalism of the quantum mechanical description of nature is fullly justified by the physical nature of things [and] that it may be connected with the principle of psycho-physical parallelism.” (Footnote 207, QTM)

This “psycho-physical parallelism” is a purely metaphysical doctrine saying that a physical process in the body is accompanied by a subjective psychological experience in the mind without any causal connection between them. Does this sound ‘New Age-y’ to you? It does to me. Yet Von Neumann not only reports Bohr’s use of this term but explicitly invokes it in his account of ‘measurement’ in quantum theory:

“..we must always divide the world into two parts, the one being the observed system, the other the observer. In the former, we can follow up all the physical processes…arbitrarily precisely. In the latter. this is meaningless. ..that this boundary can be pushed arbitrarily deeply into the interior of the body of the observer is the content of the principle of the psycho-physical parallelism.”  Von Neumann goes on to refer to the ‘ego’ of the observer as that which experiences a single outcome of the measurement, even though the physical system is described only be a set of outcomes. Connecting the two is the mysterious ‘collapse’, for which Von Neumann gives a formal representation but which he explicitly says lies outside any physically describable system.

So there you have it: the ‘ego’ of the conscious observer, in a process of ‘psycho-physical parallelism’, is seen by Quantum Physics Guru John Von Neumann as what leads to ‘collapse of the wavefunction’. This identification of the mind as a purportedly essential component of quantum phenomenology did not come from ‘New Age charlatans’; it came from the original quantum physicists.

Von Neumann was certainly not the only one. Our next visit in the trip down Quantum Memory Lane is with Nobel Laureate John Wheeler, who asserted: “no phenomenon is a real phenomenon until it is an observed phenomenon.” (“Law Without Law,” QTM, p. 183)  Wheeler coined the term ‘Participatory Anthropic Principle” (PAP), the notion that the universe is brought into existence by the participation of observers. Now, the article linked above in connection with PAP notes that Wheeler left some ambiguity about what constitutes an ‘observer’ and whether consciousness was necessary for wave function ‘collapse’. But it  also notes that Stanford University physicist Andrei Linde answers that question–whether consciousness is required–with a decisive ‘yes’. This is no so-called “New Age quack”. It is a Stanford physics professor speaking. In 2002.

Nobel Laureate Eugene Wigner also embraced consciousness as a supposedly inescapable implication of quantum theory:

“When the province of physical theory was extended to encompass microscopic phenomena, through the creation of quantum mechanics, the concept of consciousness came to the fore again: it was not possible to formulate the laws of quantum mechanics in a fully consistent way without reference to the consciousness. All that quantum mechanics purports to provide are probability connections between subsequent impressions (also called “apperceptions”) of the consciousness, and even though the dividing line between the observer, whose consciousness is being affected, and the observed physical object can be shifted towards the one or the other to a considerable degree, it cannot be eliminated. It may be premature to believe that the present philosophy of quantum mechanics will remain a permanent feature of future physical theories; it will remain remarkable, in whatever way our future concepts may develop, that the very study of the external world led to the conclusion that the content of the consciousness is an ultimate reality”  (Wigner, “Remarks on the Mind-Body Question,” Symmetries and Reflections. Indiana University Press, Bloomington, Indiana, 1967, pp.171-184.)(My emphasis)

Of course, as noted above, John Cramer and I disagree with this characterization, since TI shows that once the physical process of absorption is taken into account, there is no ‘shifty split’ of the line from physical system to consciousness of an observer. But the question as to why and how we are conscious beings is an important one that should not be disparaged, even though purely physicalist theories and approaches have a hard time accounting for it. Recent attempts to dismiss ‘metaphysics’ and ‘philosophy’ are unwarranted and unworthy of the quest for understanding of ourselves and our place in the universe. They are also basically just a repeat of the mid-20th century ‘positivist’ movement, which tried to argue that any ‘nonverifiable’ statement was ‘meaningless’. That turned out to be a fruitless and unsupportable misconception that was disposed of not long after it arose. A nice discussion of the obsoleteness of this anti-metaphysics view is given here. Responsible scientists now acknowledge that all observation is ‘laden’ with theoretical constructs including metaphysical and epistemological assumptions, and that there is no such thing as ‘objective data’ that is uncolored by such assumptions. It is simply naive to try to portray science as free of ‘philosophical musings’.

In conclusion, I’ve attempted to point out that so-called  ‘New Age Quacks’ came by their beliefs that quantum theory involves consciousness honestly: they were told this by the founders of quantum theory and they continue to hear this from highly credentialed quantum physicists. I happen to disagree that ‘consciousness’ is required to account for ‘collapse–TI shows why this is unnecessary (and see my new book for a detailed account for the general reader of why this is so). But the questions surrounding the nature of consciousness and mental processes are important ones. They should not be disparaged just because science (understood in physicalist, mechanistic terms) does not seem to have an answer.

A call for nuance in the ‘science/religion’ discussion–open letter to Dr. Sam Harris

Dear Dr. Harris:

At one point in your comments to Dr. Chopra (in “The Future of God” debate), you stated that one will never see a theoretical physicist make a categorical, dogmatic assertion. But I have often seen physicists (and philosophers of physics) do just that. One example is the often-dogmatic assertion that ‘physics implies a block world’ (i.e. that the future exists just as the past and present), when in fact physical theory does not force that conclusion (see, e.g., Raphael Sorkin’s “Relativity does not imply that the future already exists”)[1].

A famous historical example of a baldly dogmatic statement by a theoretical physicist is Niels Bohr’s categorical assertion: “There is no quantum world. There is only an abstract quantum physical description. It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature…” (as told to Aage Petersen; my emphasis)[2]. Now, Bohr had certainly done some careful philosophical investigation into quantum theory, and there are many subtleties to his thought and much that can be debated about his philosophical approach. But his explicitly dogmatic assertion about what constitutes ‘wrong thinking’ is neither scientifically nor philosophically justified. It presumes certain metaphysical and epistemological tenets that can be, and have been, rationally questioned. For example, Bohr presumed that everything about the physical world must be describable in classical terms. But in fact one can take quantum theory’s mathematical formalism as providing real information about the structure of the quantum world, even though we cannot capture that structure in the usual classical language of ordinary experience.

So unfortunately scientific researchers often deal with unresolved interpretational aspects of physical theory by lapsing into dogma. Contra Bohr, there certainly can be a quantum world, and physical theory certainly can be saying something about its nature, even if that nature is non-classical. Bohr’s assertion presupposes a conceptual and linguistic framework that is not essential or necessarily appropriate to the task at hand. In more colloquial terms, he was thinking inside a particular kind of box, and his dogmatic pronouncement sought to impose that box on all other researchers. This is why the Irish physicist John Bell railed against Bohr’s unjustified circumscription on discourse, writing a now-famous paper entitled “Speakable and Unspeakable in Quantum Mechanics”.

Of course, quantum theory certainly presents science with a wholly new kind of interpretational dilemma, since its formalism inhabits a domain mathematically larger than that of the empirical (3+1 spacetime) realm that is the traditional domain of physical science. My own proposed resolution to this dilemma involves a rationally grounded step beyond the empirical realm. That is, I offer the view that QM is a physical theory whose referent is a kind of reality that cannot be empirically observed. While this may seem radical, it should be kept in mind that Ludwig Boltzmann took a similar step when he proposed unobservable ‘atoms’ whose behavior gave rise to the macroscopic laws of thermodynamics. Though derided by his contemporaries (in particular Ernst Mach), Boltzmann’s postulation of unobservable entities turned out to be the fruitful way to go. (While we think we ‘observe’ atoms now, we actually just indirectly image them; those images are the results of particular kinds of interactions with our macroscopic instruments.)

The point here is that not all of reality describable by physical theory is necessarily contained within the empirical realm, even though physical theories can only be tested by checking against empirical data. There is no necessary logical incompatibility between these two circumstances. Indeed there are often good theoretical reasons (such as the predictive successes of theories based on unobservables) to allow for the existence of unobservables.

This naturally brings us to the topic of spirituality, which was the main subject of the debate. This feature of human experience (that is, at least of many humans) was unfortunately trivialized in the discussion by an assertion that modern monotheistic schools of thought are substantially the same as ancient worship of polytheistic gods. But in fact there are deep and significant differences between, say, Christian monotheism and worship of the Greek pantheon. I don’t think one will find a Zeus-worshiper saying things like “Show by your good life that your works are done with gentleness born of wisdom…the wisdom from above is first pure, then peaceable, gentle, willing to yield, full of mercy and good fruits, without a trace of partiality or hypocrisy…” (James 3:13, 17; New Testament)

In any case, it is perfectly possible that spiritual experience–which is primarily subjective and inward in nature–is simply a different mode of knowledge and discovery complementary to the scientific mode, the latter being primarily intersubjective and based on outer sensory experience.[3] Thus, spiritual experience might just be another way that many people ‘intuit’ the existence of sub-empirical aspects of the world. This is why it could make sense for the hermit in the cave to have an inner experience of an ‘unseen’ aspect to reality that could just possibly be the same reality described by quantum theory. Note that this suggestion is not dogma, just an offer of a possible connection between science and spirituality. In fact other researchers have suggested such a connection–e.g., Fritjof Capra (The Tao of Physics) and Gary Zukav (The Dancing Wu Li Masters). That is why I found it unfortunate to see such a schism between religion and science evidenced in the Caltech discussion, when in fact there may be some common ground between these different ways of knowing.

You and Dr. Shermer are right that organized religion has been a cause of much evil and suffering–and in fact that seemed to be your primary concern about religion in the debate. But all modes of knowledge can be abused for destructive purposes. Many wars are fought not over religion but to serve geopolitical and territorial goals, with religious or other lofty ideological principles often being used as a pretext for conquest and domination. Many scientific discoveries were ‘spinoffs’ of inquiries into how to be more effective in war (for example, the kinematics of projectile motion was a byproduct of efforts to improve cannonball aim).

Finally, to return to the concern that opened this letter: the temptation to overstate one’s case and be dogmatic is not confined to those professing claims about spiritual knowledge. It is an ever-present trap for all researchers to fall into, and those in the ‘rational’ knowledge traditions, including the sciences, are not immune[4].

Respectfully,

Ruth Kastner

P.S. Regarding free will, which you have apparently ruled out: it is certainly not a foregone conclusion that physical science disallows robust free will. That assertion, which is well on its way to becoming a dogma itself, is based either on not taking quantum theory into account or on a particular use of the quantum probability law that arguably is not justified. See, for example my earlier post:

https://transactionalinterpretation.org/2015/02/14/free-will-why-we-should-be-skeptical-of-the-skeptics/

[1] http://link.springer.com/chapter/10.1007/978-1-4020-6318-3_9

[2] The Genius of Science: A Portrait Gallery (2000) by Abraham Pais, p. 24

[3]I use the word ‘intersubjective’ rather than ‘objective’ here, because no individual scientist can get outside his or her perceptions to perceive an independently existing reality, but must rely on corroborations between many individual reports. If this seems like nitpicking, one might wish to read Chapter 1 of Bertrand Russell’s The Problems of Philosophy, which points out in graphic detail how no two people ever really see the same table, and moreover that it is a highly nontrivial question as to whether there even is a ‘real,’ objective table independently of observation.

[4] If I’m ever found to have committed such a lapse, I will readily acknowledge and correct it.

My interview with Deepak Chopra

I recently had a conversation with Deepak Chopra about the ideas in my new book, Understanding Our Unseen Reality: Solving Quantum Riddles.

A preview is here: https://www.youtube.com/watch?v=GLj3hWNGkYQ&feature=youtu.be

Full interview is here:

https://app.curiositystream.com/#/collection/162/916

What are ‘weak measurements’ and what do they tell us about quantum systems? Less than is often claimed.

The idea of ‘weak measurements’ has been much discussed recently in the popular science presses as well as in physics journals. This post aims to demystify some of the claims made about quantum systems based on weak measurements.

As illustration of the basic idea of weak measurement, consider the following analogy. A quantum shoe factory makes 2 models of shoe: a casual shoe ‘C’ and a dress shoe ‘D’. But since these are quantum shoes, the factory churns them out in a superposition of both models–call that ‘B.’ (For physicists, B is analogous to ‘spin up along x’, and C and D are spin up and down along z, respectively). It is only when Fred, the shoe checker, inspects each shoe that their nature as C or D is clearly ‘collapsed’ and thereby established. At this point, Fred places each kind of shoe in its respective bin for shipment to two different stores–one that only orders C and the other only orders D. (This is analogous to a sharp measurement that destroys the interference patern in the electron two-slit experiment.)

Now suppose it is Saturday morning, and Fred had one too many Happy Hour drinks the night before. As he ‘measures’ each shoe in its B state, his sloppiness results in some C shoes erroneously being placed in the D bin, and vice versa. If enough of these errors are made, such that Fred is just as likely to place a shoe in the wrong bin as he is to put it in the right bin, then each bin contains equal amounts of C and D, and the shoes have just had their initial combined state B confirmed. (This is analogous to having retained the interference pattern in the electron two-slit experiment.)

Suppose that hungover Fred has to sort 100 shoes. He has just enough of his faculties left to put ever-so-slightly more shoes in the correct bin than in the incorrect bin. This is the basic  ‘weak measurement’. He has almost retained the original shoe state, but not quite–the shoes in each box have gotten a bit ’tilted’ more toward C or D than they originally were.That is, each in the C bin is slightly more likely to be found in the state C than in the state D, and vice versa.

Now, to make contact with some of the claims in the literature concerning ‘weak measurements’, we have to add one more step: a follow-up careful (‘sharp’) measurement of every shoe in each of the bins. (For physicists, this is the post-selection measurement of z spin). Suppose the person carrying out this measurement is Gretchen, who unlike her co-worker Fred, did not attend Happy Hour the previous evening. Gretchen first takes bin C, and with coffee in hand, carefully measures each shoe in the box. She finds that (say) 52 of the shoes have come out (correctly) C and 48 of the shoes have come out (incorrectly) D. Then she takes the other bin  D, and finds that (say) 53 of the shoes have come out (correctly) D and 47 have come out (incorrectly) C.  At this point, only Gretchen knows and has written down which bin each shoe came from and whether Fred correctly sorted it or not. But the probability of a shoe’s having been placed in Fred’s bin C is slightly higher if it was found by Gretchen to be C.

Now, some researchers have made the following claim based on this procedure: because of Gretchen’s final measurement, each shoe somehow ‘knows’ before Fred’s sloppy sorting which of his bins it’s going to end up in. That is, Gretchen’s measurement is claimed to act rather like Merlin the Magician, who travels from the future into the past and helps beings to fulfill their destiny. The idea is that each shoe is retroactiviely steered by Gretchen’s final measurement toward its respective bin placement by the hungover Fred.

But this is incorrect, which we can see as follows. Suppose now a shoe store representative, Helen, comes to the factory just after Gretchen’s quality control. Helen decides to play a guessing game with her, as follows. She picks up each C or D shoe  and tries to guess in which bin Fred had put it. For a shoe that ended up C, she has slightly better luck guessing  that it came from Fred’s bin C, and similarly with D.Why is this? Simply because the result that Gretchen found was more likely to have come from a state favoring that outcome (a state created by Fred’s sloppy measurement) than from a state inhibiting that outcome. We don’t in fact need Merlin the Magician to explain any of this. The situation is no different conceptually from being able to predict that a person coming to the U.S. from Poland is more likely to be of Polish ancestry than, say, Japanese ancestry. The fact that we now see that the person came from Poland does not retroactively cause the person to have been born of Polish parents!

So the next time you see claims such as  “future measurements affect past measurement results,” be wary. None of the quantum shoes sorted by Fred were subject to a Merlin-like retrocausal influence from Gretchen. Fred simply tilted the shoes to  states more likely to end up with one property, upon measurement, than the other. And this is all that is demonstrated by these kinds of experiments: standard quantum mechanics.

My interview with Zain Khan, Glow TV

A brief introduction to the puzzles of quantum theory and why we should try to understand its message about reality:

Dr. Quantum on my new book (http://www.worldscientific.com/worldscibooks/10.1142/p993)

Fred Alan Wolf (aka ‘Dr. Quantum’) is a best-selling author of philosophically-oriented books on modern physics  for the general reader. He recently offered this review of my new book, Understanding Our Unseen Reality: Solving Quantum Riddles (Imperial College Press):

Ruth Kastner is emerging as one of the latest new interpreters of the mysteries of quantum physics and as such provides a unique perspective that she calls the Possibilist Transactional Interpretation or PTI based on the TI theory of John Cramer, expounded earlier by Paul Davies, with both theories based on Richard P. Feynman’s earlier absorber theory.  In her latest book she expands on her PTI through using some clever everyday analogies to bring the complexities of quantum physics into the realm of the non-expert. I think that she handles the non-relativistic quantum physics quite well with the PTI, but I find it difficult extending the PTI into quantum field theory although she does make a clear distinction between virtual and real processes using it. For example, there is little discussion of how antiparticles are related to their mirror image particles using the PTI—something that Feynman’s concept of a particle going backward in time with negative energy appearing as an antiparticle going forward in time with positive energy handles quite well.  This is certainly a well-worthwhile read for those of you interested in how we are still grappling with understanding quantum physics 115 years after its inception.”-Fred Alan Wolf

I thank Dr. Quantum for his review and for his question concerning antiparticles. PTI has no problem handling antiparticles, but I decided to hold off on that topic and to address it in a future work. For this introductory book, I wanted to focus on the basics. But in a nutshell, for those who are curious, there are four solutions to the Dirac equation and two of those involve negative energies. Those are still antiparticle states (i.e., positrons) in PTI, and they end up conveying positive energy in any transaction. Remember also that at the virtual or offer-wave level, we are dealing only with possible energy, and this is not restricted to positive values. It is only actualized energy that must be positive. I welcome discussion concerning this or any other aspect of PTI, via this website.

I should add that I neglected to reference Prof. Wolf’s book Star Wave (1984, Macmillan) which was the first book for the general reader to discuss Cramer’s TI. I apologize for this oversight. It should be noted however that his interpretation of the transactional picture (invoking consciousness to explain collapse) differs from PTI in that PTI takes the physical process of absorption, together with a form of spontaneous symmetry breaking, as sufficient to explain the fact that we find deterninate measurement results. This issue is also discussed in Chapter 4 of my earlier book.

A Unified Account of Relativistic and Non-Relativistic Quantum Theory

The non-technical presentation of this material is available in my new book for the general reader, available here and as a Kindle version here.

The Possibilist Transactional Interpretation:

A Unified Account of Relativistic and Non-Relativistic Quantum Theory

that Solves the Problem of Measurement

Ruth E. Kastner

17 February 2015

1 Introduction

The transactional interpretation of quantum mechanics (TI) was initially proposed by John G. Cramer in the 1980s. His most comprehensive exposition is found in Cramer (1986). This time-symmetric interpretation of quantum mechanics gives rise to a physical basis for the Born Rule for the probability of an event. The Born Rule specifies that the probability of an outcome is given by the square of the wave function corresponding to that outcome.

TI was inspired by the Wheeler-Feynman (WF) time-symmetric or ‘direct action’ theory of classical electrodynamics (Wheeler and Feynman 1945, 1949). In the WF theory, radiation is a time-symmetric process. A charge emits a field in the form of half-retarded, half-advanced solutions to the wave equation; the response of absorbers then gives rise to a radiative process that transfers energy from an emitter to an absorber. In this picture, the usual quantum state is called an ‘offer wave’ (OW) and the advanced response from the absorber is called the ‘confirmation wave’ (CW). In terms of state vectors, the OW is represented by a ket, |Y>, and the CW by a dual state vector or ‘brac,’ <F|.

The basic transactional picture has been extended by this author, in a ‘possibilist’ ontology, to the relativistic domain. This version of the interpretation is referred to as ‘Possibilist Transactional Interpretation’ or PTI (Kastner 2012). A theoretical basis can be found in the Davies application of the direct-action picture of fields to quantum electrodynamics (Davies 1971, 1972). However, PTI departs from the Davies treatment in two ways: (i) virtual particles are clearly distinguished from real particles (see also Kastner 2014a) and (ii) the coupling amplitude is identified as the amplitude for generation of an offer or confirmation wave, in a transactional account. These developments allow the interpretation to provide a smooth transition between the non-relativistic and the relativistic domains. The latter can be viewed as the ‘birthplace’ of offer waves, in a physically well-defined (although fundamentally stochastic) manner. These points will be discussed in Sections 3 and 4.

It should also be noted that the direct-action theory of fields provides an elegant and effective escape from Haag’s Theorem, a famously vexing result showing that the interaction picture of the standard quantized fields does not exist. This point is discussed in Kastner (2015).

  1. How TI explains von Neumann’s Measurement Process and the Born Rule

John von Neumann formulated in precise terms an account of measurement which, despite its practical utility, remains mysterious in any interpretation except TI. To review, von Neumann delineated two different processes that take place in quantum systems. The second of these, which he called ‘Process 2’, is the unitary evolution described by the Schrödinger Equation. The first of these, called ‘Process 1’, is the transition of a quantum system from a pure state to a mixed state upon measurement, i.e.:

von Neumann process 1

The coefficients |cn|2 are the probabilities given by the Born Rule for each of the outcomes yn.

Von Neumann noted that this transformation is acausal , nonunitary, and irreversible, yet he was unable to explain it in physical terms. He himself spoke of this transition as dependent on an observing consciousness. However, we need not view the measurement process as observer-dependent. If we take into account the advanced responses of absorbers, then for an OW described by |Y>, we have for a collection of numbered absorbers:

 von Neumann Figure

 In the above diagram, an initial offer wave from emitter E passes through some measuring apparatus that separates it into components <yn|Y> |y n>, each reaching a different absorber n. Each absorber responds with an advanced (adjoint) confirmation <yn| <Y|y n>. In TI, these OW/CW encounters are called incipient transactions. They are described in probabilistic terms by the product of the OW and CW, which gives a weighted projection operator: <yn|Y><Y|yn> |y n><y n | = |cn|2 |y n><y n |. If we add all the incipient transactions, we clearly have the density operator representation of ‘Process 1”.

Thus, by including the advanced responses of absorbers, we have a physical account of measurement as well as a natural explanation of the Born Rule and Von Neumann’s ‘Process 1’. The response of absorbers is what creates the irreversible act of measurement and breaks the linearity of the basic deterministic propagation of the quantum state. Since the conserved physical quantities can only be delivered to one absorber, there is an indeterministic collapse into one of the outcomes yk with a probability given by the weight |ck|2 of the associated projection operator |y k><y k |. This is called an actualized transaction, and it consists in the delivery of energy, momentum, angular momentum, etc., to absorber k. That absorber that figuratively wins the incipient transaction ‘lottery’ is called the ‘receiving absorber’ in PTI. The process of collapse precipitated in this way by absorber response(s) can be understood as a form of spontaneous symmetry breaking (this is discussed in Chapter 4 of Kastner 2012a).

  1. The Possibilist Ontology and why it is necessary

If the system under consideration consists only of a single quantum, its associated state vector has only 3 spatial degrees of freedom. In that case, one can think of the wave function as inhabiting spacetime. However, any composite system of N quanta has 3N spatial degrees of freedom, and therefore cannot be considered a spacetime object. Any realist interpretation of the quantum state must take this fact into account. As a realist interpretation, the transactional picture describes real objects that inhabit a realm described not by 3+1 spacetime dimensions but by the 3N dimensions of the relevant Hilbert space (in addition to any spin degrees of freedom).             If we think of the spacetime realm as the realm of concrete, actualized events, then the quantum entities described by state vectors must have a different ontological status. In PTI they are viewed as physical possibilities. The latter are precursors to any actualized spacetime event. In view of quantum indeterminism, they are necessary but not sufficient conditions for observable events. The necessary condition is the emission of an offer wave and confirming response(s) to that offer wave from one or more absorbers. This sets up a set of incipient transactions as described in the previous section. At that point we have non-unitary collapse, and only one of the set is actualized to become a spacetime interval with well-defined emission and absorption endpoints—that is the sufficient condition. Thus, not all OW/CW exchanges will result in actualized transactions corresponding to spacetime events.

  1. The relativistic domain as the birthplace of offer waves

Nonrelativistic quantum mechanics describes a constant number of particles emitted at some locus and absorbed at another. However, in the relativistic domain we are dealing with interactions among various coupling fields, and the number and types of quanta are generally in great flux. Such interactions are described in terms of scattering. The internal connections –i.e., virtual particles– are characterized at each scattering vertex by the coupling amplitude (in the case of QED, the elementary charge e) and are neither offers nor confirmations according to relativistic PTI. These are ‘internal lines’ in which the direction of propagation is undefined; i.e., there is no fact of the matter concerning which current ‘emitted’ and which ‘absorbed’ the virtual quantum. In that case, as shown by Davies (1971), the Feynman propagator DF of standard QED can be replaced by the time-symmetric propagator in the direct-action theory.

According to PTI, the field coupling amplitudes, which are not present in the non-relativistic case, represent the amplitude for an offer or confirmation to be generated. This is a feature of the interpretation appearing only at the relativistic level, in which the number and type of particles can change. It is a natural step, in view of the fact that in standard QED the coupling amplitude is the amplitude for a real photon to be emitted. In PTI, a ‘real photon’ corresponds to an offer wave.[1] So we can think of virtual particles as necessary but not sufficient conditions for real particles, which correspond to offer waves in PTI. We see that the relativistic level brings with it a subtler form of the uncertainty associated with the nonrelativistic form of the theory, i.e., the amplitudes of quantum states themselves.

Thus, in PTI we have a clear distinction between virtual and real particles: virtual particles are not offer waves. Rather, they are precursors to offers or confirmations that do not rise to that level. In general, they are not on the mass shell. In the direct-action theory underlying the transactional picture, they are represented by time-symmetric propagators, not Fock space states. On the other hand, real particles correspond to offer waves, which are on the mass shell and are represented by Fock space states. In the direct action picture, offer waves are always responded to by confirmations, so the term ‘real photon’ (in the context of QED) can refer either to a Fock space state |k> (i.e., an OW) or to the projection operator |k><k| (where k is 3-momentum), depending on the context.

One of the criticisms of the original Cramer theory was that it took emitters and absorbers as primitive. PTI overcomes this limitation by providing well-defined physical conditions for the generation of OW and CW: both have their origins in the incessant virtual particle activity that is the coupling between fields, such as between the Dirac field and the electromagnetic field. When the interacting photons are on the mass shell and satisfy energy conservation, they are capable of achieving OW status, which in turn generates a CW response from eligible interacting fermionic currents. This is an inherently indeterministic process, since it is described by two factors of the coupling amplitude—i.e., the fine structure constant for QED (as well as the square of the relevant transition amplitude between the initial and final states, with satisfaction of energy conservation). The two factors of the coupling amplitude correspond to the two vertices that are involved in each scattering interaction.

Details of the conditions for the emission of an offer wave instead of an internal process (i.e. a propagator linking two vertices) are given in Kastner (2014a), along with a discussion of how this picture can shed light on the computations necessary to obtain atomic decay rates. The latter are spontaneous emission processes in which virtual photon exchange is spontaneously elevated to a real photon offer and confirmation. Such elevations must of course satisfy energy conservation. That is the only way a would-be virtual photon becomes a real photon—only those photons satisfying the mass shell condition and the energy conservation requirements are eligible to be elevated in this way.

One way to visualize the new level of uncertainty presented in the relativistic domain is in terms of a coin flip. In the nonrelativistic theory, the coin flip has two outcomes: (1) an offer wave |Y> is emitted; (2) no offer wave is emitted. So we either have an offer wave or we don’t. Of course, if we do have an offer wave, it is characterized by its amplitudes <x|Y> for reaching various absorbers X. (If we have an offer wave, we also have confirming responses <x| to that offer, so the two always go hand-in-hand.) This is what made the notion of emitter and absorber in the original TI seem primitive or arbitrary: at what point do we call something an emitter or absorber? However, this problem is remedied at the relativistic level. At that level it is as if the coin becomes much thicker, and a third option becomes available: the coin lands on its side. This in-between, ‘sideways’ outcome corresponds to the virtual quantum. This result has a well-defined probability to yield an emitted offer and confirming response(s) (subject also to the satisfaction of conservation laws as noted above). That probability is the square of the coupling amplitude—i.e., the fine structure constant for QED. So we have two levels of probability amplitudes: (i) the usual, nonrelativistic amplitude of a quantum state <x|Y> corresponding to a particular outcome X; and (ii) the relativistic amplitude for a virtual quantum to become a real quantum representable by a quantum state |Y>.

  1. Spacetime as the growing set of actualized events

It is often claimed that relativity implies a block world – that is, an ever-present spacetime in which all past, present and future events exist in an equally robust sense. The main argument used in support of that claim is termed ‘chronogeometrical fatalism’ (cf. Stein 1991, pp. 148-9).   However, that argument rests on certain assumptions, such as the ontological status of ‘lines of simultaneity’, and a substantival view of spacetime as a ‘container’ for events, that do not necessarily hold. (See Kastner 2012a, §8.1.4 for a rebuttal of chronogeometrical fatalism. See also Sorkin (2007) for a rebuttal to this common but erroneous assumption that a ‘block world’ is necessarily implied by relatibity.)

A different model, of a growing spacetime, is perfectly viable; one such model is the ‘causal set’ approach as proposed by Bombelli et al (1987) (see also Sorkin (2003) and Marolf and Sorkin (2006)). A causal set (causet) C is a locally finite partially ordered set of elements on which is defined a binary relation < . The following properties apply to the causet:

(i) transitivity: (∀x, y, z ∈ C)(x ≺ y ≺ z ⇒ x ≺ z)

(ii) irreflexivity: (∀x ∈ C)(x ~< x)

(iii) local finiteness: (∀x, z ∈ C) (cardinality {y ∈ C | x ≺ y ≺ z} < ∞)

Properties (i) and (ii) imply that the elements are acyclic, while (iii) dictates that the elements form a discrete rather than continuous set. The result is a well-defined causal order of distinct events that can be associated with a spacetime manifold possessing a temporal direction defined by that causal order. As new elements are introduced into the causet, its growth describes a growing, temporally unidirectional spacetime.

The actualized transactions of PTI can readily function as the dynamics underlying the process of ‘sprinkling’ new events into the causal set that is the spacetime manifold. An introduction to the basic concepts involved can be found in Kastner (2014b). The addition of new events to the spacetime causal set must be a Poissonian process in order to preserve relativistic covariance. Interestingly, the generation of offer waves is just such a process, since (as noted in the previous section) these are governed by Poissonian decay rates for the bound states, such as atoms, that give rise to those offer waves.

Offers are always responded to by confirmations, one of which will result in the actualization of a transaction that defines a spacetime interval in terms of its corresponding emission and absorption event. The absorption event of such an actualized transaction defines the present for that receiving absorber, while the emission event is actualized in the timelike (or lightlike) past relative to that absorption. The transferred quantum of energy constitutes a link between the two events that establishes their temporal relationship. It is this linking process between actualized events that establishes spacetime intervals and provides a clearly defined structure of the spacetime ‘fabric’, including its naturally oriented temporal direction towards the unactualized future. In this picture, the present is a strictly local property and cannot be extended along a ‘line of simultaneity’ into the spacelike ‘elsewhere’. Rather, events that are unrelated by the partial ordering of the transactional links have no temporal relationship.

  1. Conclusion

This article has reviewed the essential concepts of the ‘Possibilist Transactional Interpretation’ (PTI) of quantum theory. It has been argued that PTI can provide a solution to the notorious problem of measurement, as well as providing a unified account of nonrelativistic and relativistic quantum theory. At the nonrelativistic level, we deal only with pre-existent offer waves |Y> and their confirmations from one or more absorbers indexed by X, which correspond to dual states <Y|x> <x|. The confirming response constitutes the onset of the measurement process, where the collapse to a particular outcome |x><x| concludes the measurement process.       The collapse is not a process that occurs within spacetime, and that is why it has been so notoriously difficult to give a spacetime account of collapse (cf. Aharonov and Albert (1981) and Kastner (2012, §6.7). Rather, collapse corresponds to the creation of spacetime events from the quantum substratum. That substratum is composed of physical possibilities described by quantum states and of the virtual processes, described by time-symmetric propagators, which are the precursors to those states. The two events created via an actualized transaction are (i) the emission and (ii) the absorption of real energy. That transfer of real energy from the emitter to the receiving absorber defines a spacetime interval and a temporal direction, the emitter defining the past and the absorption defining the present for that absorber. Thus we gain deep physical meaning corresponding to the mathematical fact that energy is the generator of time translation.

At the relativistic level, we take into account virtual processes and their couplings with candidate emitters and absorbers. Such virtual processes are necessary but not sufficient conditions for the generation of offers and confirmations. The latter occur on a stochastic basis, where the probabilities for their occurrence correspond to decay rates. Thus, while the transactional picture of measurement involves objective uncertainty, that uncertainty is precisely quantifiable both at the nonrelativistic and relativistic levels..

References

Aharonov, Y. and Albert, D. (1981) . “Can we make sense out of the measurement process in relativistic quantum mechanics?” Physical Review D 24, 359- 370.

Bombelli, L., J. Lee, D. Meyer and R.D. Sorkin (1987). “Spacetime as a causal set”, Phys. Rev. Lett. 59: 521-524.

Cramer J. G. (1986). “The Transactional Interpretation of Quantum Mechanics.” Reviews of Modern Physics 58, 647-688.

Davies, P. C. W. (1971).”Extension of Wheeler-Feynman Quantum Theory to the Relativistic Domain I. Scattering Processes,” J. Phys. A: Gen. Phys. 6, 836

Davies, P. C. W. (1972).”Extension of Wheeler-Feynman Quantum Theory to the Relativistic Domain II. Emission Processes,” J. Phys. A: Gen. Phys. 5, 1025-1036.

Kastner, R. E. (2012). The Transactional Interpretation of Quantum Mechanics: The Reality of Possibility. Cambridge University Press.

Kastner, R. E. (2014a). “On Real and Virtual Photons in the Davies Theory of Time-Symmetric Quantum Electrodynamics,” Electronic Journal of Theoretical Physics 11, 75–86. Preprint version: http://arxiv.org/abs/1312.4007

Kastner, R. E. (2014b). “The Emergence of Spacetime: Transactions and Causal Sets,” to appear in Ignazio Licata, ed., The Algebraic Way. Springer.

Kastner, R. E. (2015). “Haag’s Theorem as a Reason to Reconsider Direct-Action Theories,” to appear in International Journal of Quantum Foundations. Preprint version: http://arxiv.org/abs/1502.03814

Marolf, D. and R.D. Sorkin,”Geometry from order: causal sets ” in: Einstein Online Vol. 02 (2006), 1007.

Sorkin, R. D. (2003) “Causal Sets: Discrete Gravity (Notes for the Valdivia Summer School),” In Proceedings of the Valdivia Summer School, edited by A. Gomberoff

Sorkin, R. D. (2007). “Relativity theory does not imply that the future already exists: a counterexample,” in Vesselin Petkov (editor), Relativity and the Dimensionality of the World. Springer. Preprint version: http://arxiv.org/abs/gr-qc/0703098

Wheeler, J.A. and R. P. Feynman, “Interaction with the Absorber as the Mechanism of Radiation,” Reviews of Modern Physics, 17, 157–161 (1945)

Wheeler, J.A. and R. P. Feynman, “Classical Electrodynamics in Terms of Direct Interparticle Action,” Reviews of Modern Physics, 21, 425–433 (1949).

[1] The term ‘real photon’ can also be applied to an actualized transaction, depending on the context. The term is really a conflation of two different physical situations in the standard approach which can be clearly disambiguated in the transactional approach.

Free Will Part II: No need to be disillusioned

Last week I argued that agents making free choices do not in fact have to violate any physical law, in view of quantum indeterminism. Rather than being a ‘slave’ to the quantum statistics, as some philosophers have argued (e.g. Ted Sider, 2005), a choosing agent can be governed by quantum propensities while still having enough ‘wiggle room’ to make free choices—choices that are fundamentally caused by the agent’s volitional powers.

To review, the Born Rule for probabilities of quantum outcomes can only be violated if it is applied to a well-defined quantum state in which many precisely repeated trials yield a distribution of outcomes that deviate significantly and reliably from the rule (where the frequencies of the outcomes represent the probabilities). The ‘wiggle room’ is available to the human being because, as a complex biological system, he or she is not described by a well-defined quantum state subject to a well-defined measurement observable over a time interval long enough to generate a valid statistical application of the rule.  The Born Rule may still govern the agent’s choices at each instant, but no deviation from the rule can be established if the agent’s physical state (at the quantum level) is continually changing in an unpredictable and uncontrollable way. Thus, there is no necessary violation of the quantum statistics on the part of a freely choosing macroscopic biological system like a human being.

This week, I examine a view known as ‘Disillusionism’. This approach says that free will is an illusion, yet people can have meaningful and fulfilling lives without free will. Such an approach has recently been advocated by philosopher Greg Caruso (2013). Disillusionism is based either on a deterministic interpretation of quantum theory (such as the Bohmian interpretation), or on taking the quantum statistics as constraining our choices and actions so tightly that in effect those choices are pre-determined. As I have argued, I think the latter is based on a misunderstanding of the quantum statistics and the circumstances required for their application. So let us suppose that philosophers advocating Disillusionism are doing so because they think that (despite quantum theory) all actions are truly predetermined, and/or that we live in a ‘block world’ where all past, present and future events ‘already’ exist.

If all actions are predetermined, physically we are akin to dominoes that are being figuratively ‘fallen on’ by other dominoes. Each time that happens, whether or not we also will fall depends not on anyone’s choice, but simply on the physical conditions of each fall. For example (figuratively speaking), sometimes one domino will fall on another, but the neighboring domino will not be knocked over, simply because the first domino was not quite close enough to the second one to overcome its inertia.

In this picture, all our choices and actions are determined by circumstances and forces over which we have no control at all. Whenever we do anything, it is because we are compelled to do so. If one doesn’t like the term ‘compelled,’ perhaps another word is ‘propelled.’ Whatever words we use to describe the situation, we are effectively automatons in which each input results in a single fully predictable and unavoidable output. This means that whenever we perceive ourselves as ‘trying’ to do something, it is in fact already decided whether our ‘attempted’ action will occur, and what its outcome will be. Therefore, in this disillusionist approach, isn’t our subjective sense of ‘trying’ to do things also an illusion that would need to be rejected?

Suppose dominoes were sentient. While they might be able to perceive themselves as being involved in various processes and as exerting effort, in fact they are not self-propelled. Instead, they are propelled by forces beyond their control, since all their actions are fully dictated by those forces. So, in what sense is any of those dominoes really ‘trying’ to do anything? Every action that occurs is fully explained by physical processes and forces, so no ‘trying’ on the part of any of the dominoes is really part of the explanation for anything that occurs. If a domino perceives itself as exerting an effort, that perception must be just a byproduct of the actions in which he is fully determined by forces beyond his control to engage, and therefore just another aspect of the free will illusion. Without free will, ‘trying’ is superfluous, and any conscious entity is simply a sentient automaton.

The point of the above is that we can’t have it both ways: either (1) we have free will, in which case we can exert creative efforts through our own volitional capacity toward specific aims that we are trying to achieve, or (2) under disillusionism, we are simply automatons that don’t actually try to do anything. We just fall, as dominoes, where we are propelled to fall, and our subjective perceptions that we are exerting creative efforts are just as illusory as our subjective sense that we have free will. Thus, it is doubtful that disillusionism about free will can be consistent with a meaningful, creative life. Without free will, each person is an automated cog in a machine—even if perhaps a sentient one.

However, ‘disillusionism’ is certainly not demanded by physical law, as I pointed out in Part I. We can indeed be self-propelled, and although we certainly are subject to some forces beyond our control, we need not see ourselves as primarily propelled by them. The effort we must exert to accomplish our chosen tasks could be just as real as our ability to make those choices.

References

Caruso, Gregg (2013). Free Will and Consciousness: A Determinist Account of the Illusion of Free Will. Lexington Books.

Sider, T. (2005). “Free Will and Determinism,” in Riddles of Existence, by Earl Conee and Theodore Sider (Oxford: Clarendon Press), pp. 112-133.

Free Will: Why We Should be Skeptical of the Skeptics

It has become quite popular lately to view the notion of free will as a misconception to be ‘debunked.’ To be sure, if we really do not have free will, we should be prepared to face that fact. But is it really a fact? I will argue here that key arguments against robust free will are seriously overrated. These are:

(1) Physical theory implies a block world (i.e. all events exist, including future events).

(2) Physical law, including indeterministic quantum physics, is inconsistent with free will.

Concerning (1), this very widespread misconception has been refuted in the literature. See, e.g., Sorkin (2007) and Kastner (2012), Chapter 8.

Concerning (2), it seems clear that if the world were fully deterministic, then all our actions would be fully determined by prior causes, so in that case there would be no room for a robust form of free will. (Caveat: there is an approach called ‘compatibilism’ that argues that free will is compatible with determinism; I find this approach decidedly unconvincing, but it’s something readers can look into.) However, in most interpretations, quantum theory implies that the world is genuinely indeterministic: given well-defined conditions, it is impossible to predict with certainty what will follow from those conditions. Nevertheless, free will skeptics such as Ted Sider (2005) have argued that free will must violate even the statistical laws of quantum theory. His argument basically assumes that a free agent, considered as a quantum system, would make choices that would violate the quantum statistical laws applying to the outcomes of his actions.

There are two serious problems with this argument. First, as noted by Clarke (2010),

“probabilistic laws of nature also do not require, for any finite number of trials, any precise distribution of outcomes. The probabilities involved…are the chances that events of one type will cause, or will be followed by, events of another type…These probabilities, we may assume, determine single-case, objective probabilities, or propensities. Actual distributions can diverge from proportions matching these probabilities.”

Thus, a statistical law is not ‘violated’ unless very large numbers of precisely repeated experimental runs yield statistically significant deviations from expected mean values, where even ‘statistically significant’ can be a matter of context and degree. Highly unlikely strings of outcomes may occur, and yet a statistical law may still not be violated. The point here is that the demonstration of a real violation of a statistical law requires a very high hurdle of empirical evidence.

The second problem is in trying to apply the quantum statistical law – the Born Rule – to human agents, which are macroscopic biological systems. In order to predict empirically useful probabilities of outcomes with the Born Rule, one must have a clearly defined system and a clearly defined observable being measured on that system. A definition of a system must specify how many degrees of freedom (usually considered as ‘particles’) are in play, and exactly what the initial state of that system is. A definition of an observable must specify exactly what forces are acting on the system and what sort of ‘detection’ constitutes each outcome of the observable being measured. These requirements may be straightforwardly met for microscopic systems in the laboratory, but it is a highly nontrivial matter as to whether they may be met under conditions obtaining in the context of human behavior.

Sider essentially argues that a human agent governed by the Born Rule should be able to make choices that would observably deviate from the Born Rule. But this assumes that one could set up repeatable experiments in which the agent could be precisely defined as a ‘quantum system’ whose applicable observable was so tightly defined as to allow detection of such deviations. It is only if such deviations were in principle detectable that there could be a violation from the statistical laws of quantum mechanics, as observed in Clarke’s remark quoted above. However, there are very good reasons to think that this is not the case.

For one thing, as noted above, one has to be able to perform precisely repeatable experiments. Does exposing a given human agent to repeated opportunities to make a choice constitute a precisely repeatable experiment of this type? Why should we think so? The human agent is an open system, continually exposed to variable influences from his or her environment: air currents, radiant energy, etc; as well to internal fluctuations (number of blood cells in the brain, number of activated neurons, etc.). Assuming the brain is the most relevant bodily system concerning the choice, the state(s) and the number of relevant degrees of freedom in the brain are in continual flux. No matter how tightly one might attempt to control the agent’s environment, one is dealing with an enormously sensitive, complex and ill-defined system, from a quantum-mechanical perspective.

At the level of individual instances, the Born Rule gives only propensities for outcomes. A human agent might instantaneously be subject to those propensities; yet, given quantum indeterminism, could still have room to make a free choice– one that would not violate any statistical law. This is because another instance outwardly presenting the same choice to the agent is in fact highly unlikely to constitute an identical repetition of the relevant initial conditions: i.e., the agent is almost certainly not in exactly the same state that he or she was just prior to the previous choice. Therefore, the Born Rule propensities are likely not really the same as in the previous instance. Even if the experiment is repeated many times, a resulting set of outcomes in which so many parameters are ill-defined and subject to change cannot be used to determine whether a statistical law is being violated.

Thus, it is a highly nontrivial matter to try to apply the Born Rule to macroscopic biological systems; yet claim (2) presumes without argument that one can straightforwardly do so. If this is not in principle possible due to the intrinsically ill-defined and/or ever-changing nature of the macroscopic physical system constituting the choosing agent, then there is no necessary violation of the Born Rule. This is so even if the agent’s choices are governed by the Born Rule, in terms of propensities, for each individual instance.

The bottom line: rather than see quantum theory as falling under yet under ‘physical law’ that is supposedly violated by free will, we can view quantum theory as being precisely the kind of physical law that allows for free will.

References

Clarke, R. (2010). “Are we free to obey the laws?”, American Philosophical Quarterly 47, pp. 389-401

Kastner, R. E. (2012). The Transactional Interpretation of Quantum Mechanics: The Reality of Possibility. Cambridge University Press.

Sider, T. (2005). “Free Will and Determinism,” in Riddles of Existence, by Earl Conee and Theodore Sider (Oxford: Clarendon Press), pp. 112-133.

Sorkin, R. D. (2007). “Relativity theory does not imply that the future already exists: a counterexample,” in Vesselin Petkov (editor), Relativity and the Dimensionality of the World. Springer. Preprint version: http://arxiv.org/abs/gr-qc/0703098

[1] The approach known as ‘compatibilism’ holds that free will is compatible with determinism. I will not address that here. However, I do think that compatibilism yields a very impoverished notion of free will.