**Adlam, Emily**

*Two Roads to Retrocausality*

I will discuss two distinct conceptions of retrocausality and argue that `all-at-once’ retrocausality is more coherent than the alternative dynamical picture. I will then argue that since the all-at-once approach requires probabilities to be assigned to entire histories or mosaics, locality is somewhat redundant within this picture. I will consider some possible motivations for using dynamical retrocausality to rescue locality, arguing that none of these motivations is adequate, and finally I will show that accepting the existence of nonlocality and insisting on the nonexistence of preferred reference frames leads naturally to the acceptance of all-at-once retrocausality.

**Argaman, Nathan**

*Toy models for quantum entanglement — statistical independence and the arrow of time*

Bell’s theorem is a no-go mathematical theorem, so one of its assumptions (at least) must be violated in any physical theory which reproduces the success of Quantum Mechanics. These assumptions are reviewed, with special attention to (no) superdeterminism and (no) retrocausality, the major topics of this workshop. In particular, Bell’s locality assumption is generalized into a time-symmetric locality assumption called “Continuous Action,” and toy models which fulfill this assumption are discussed. These models violate the time-asymmetric “No Future-Input Dependence” condition, i.e., the no retrocausality assumption, but still conform to the signal causality condition which is required for consistency (they are not completely time symmetric). This situation is closely analogous with the tension between signal locality and quantum nonlocality in standard descriptions of quantum entanglement.

Reference: K.B. Wharton and N. Argaman, “*Colloquium*: Bell’s theorem and locally mediated reformulations of quantum mechanics,” Rev. Mod. Phys. **92**, 021002 (2020).

**Bradshaw, Peter**

*Towards an Algebraic Theory of Spin*

The discovery of spin in the twentieth century is widely regarded as one of the many heralds of the quantum age in physics. It presents as an intrinsic angular momentum of a particle for which there is no classical analogue. It demonstrates the ability to couple with other forms of angular momentum, the physical consequences of which are myriad and well-known. It has also long been established that the physically allowed spins conform to (ir-)reducible representations of the universal covering group of the “homogeneous” space(-time) symmetries.

However, what has remained somewhat mysterious is how necessary quantum physics is to this story. In particular, what features of quantum theory are required for the complete description of spin as we know it? To begin to answer this question, spin as we know it must be deconstructed. We will present the established non-relativistic theory of spin and distill the essential components from the formalism. In so doing we will motivate the characterisation of spin by abstract algebraic means. This method reveals striking connections to the Clifford and Kemmer algebras, and forms the core of a new, first-principles approach to the representation theory.

**Catani, Lorenzo**

*A mathematical framework for operational fine tunings*

In the framework of ontological models, the inherently nonclassical features of quantum theory always seem to involve properties that are fine tuned, i.e. properties that hold at the operational level but break at the ontological level. Their appearance at the operational level is due to unexplained special choices of the ontological parameters, which is what we mean by a fine tuning. Famous examples of such features are contextuality and nonlocality. In this article, we develop a theory-independent mathematical framework for characterizing operational fine tunings. These are distinct from causal fine tunings — already introduced by Wood and Spekkens in [NJP,17 033002(2015)] — as the definition of an operational fine tuning does not involve any assumptions about the underlying causal structure. We show how known examples of operational fine tunings, such as Spekkens’ generalized contextuality, violation of parameter independence in Bell experiment, and ontological time asymmetry, fit into our framework. We discuss the possibility of finding new fine tunings and we use the framework to shed new light on the relation between nonlocality and generalized contextuality. Although nonlocality has often been argued to be a form of contextuality, this is only true when nonlocality consists of a violation of parameter independence. We formulate our framework also in the language of category theory using the concept of functors.

**Cohen, Eliahu**

*Superdeterminism and Retrocausality within the Two-State-Vector Formalism*

The Two-State-Vector Formalism (TSVF) of quantum mechanics emerged as an attempt to symmetrize the measurement process and specifically the “reduction of the wave packet” which seems to introduce irreversibility into quantum theory. While its predictions have been identical to those of textbook quantum mechanics, the TSVF managed over the years to address not only the measurement problem [1], but also many kinematically and dynamically nonlocal phenomena, counterfactuals [2], realism and energetic causal sets [3], emergence [4] and even the black-hole information paradox [5]. I would therefore like to discuss this formalism, mostly focusing on its recent formulations [6,7]. I will make connections between the TSVF and related approaches, trying to define the unique sense in which it is “retrocausal” and “superdeterministic”.

[1] Y. Aharonov, E. Cohen, T. Landsberger, “The Two-Time Interpretation and Macroscopic Time-Reversibility”, Entropy 19, 111 (2017).

[2] Y. Aharonov, E. Cohen, S. Popescu, “A Dynamical Quantum Cheshire Cat effect and Implications for Counterfactual Communication”, Nat. Commun. 12, 4770 (2021).

[3] E. Cohen, M. Cortes, A.C. Elitzur, L. Smolin L., “Realism and causality I: Pilot wave and retrocausal models as possible facilitators”, Phys. Rev. D 102, 124027 (2020), “Realism and causality II: Retrocausality in Energetic Causal Sets”, Phys. Rev. D 102, 124028 (2020).

[4] Y. Aharonov, E. Cohen, J. Tollaksen, “Completely top–down hierarchical structure in quantum mechanics”, Proc. Natl. Acad. Sci. USA 115, 11730-11735 (2018).

[5] E. Cohen, M. Nowakowski, “Comment on ‘Measurements without probabilities in the final state proposal’”, Phys. Rev. D 97, 088501 (2018).

[6] Y. Aharonov, E. Cohen, F. Colombo, T. Landsberger, I. Sabadini, D.C. Struppa, J. Tollaksen, “Finally Making Sense of the Double-Slit Experiment”, Proc. Natl. Acad. Sci. USA 114, 6480-6485 (2017).

[7] M. Waegell, E. Cohen, A.C. Elitzur, J. Tollaksen, Y. Aharonov, “Quantum reality with negative-mass particles”, arXiv:2201.09510.

**Donadi, Sandro***A Superdeterministic Toy Model*

The violation of Bell’s inequalities, observed in several experiments, tell us that at least one of the two assumptions taken in Bell’s theorem, namely Bell’s factorisability (or Bell’s locality) and statistical independence, is not true. Typically, statistical independence is supposed to hold true, which leads to the conclusion that the first hypothesis must be violated i.e. Nature is non-local. Here we focus on the second possibility, that is we assume that statistical independence is violated i.e. we consider superdeterministic models. We discuss this possibility focusing on a specific toy model where the hidden variables are correlated to the future setting of the measurement device. We discuss the main features and limitations of this model and why it is not fine tuned. We then conclude by giving a perspective on possible generalization of the toy model to more realistic ones.

**Flack, Robert**

*Using the STIRAP process to observe the quantum potential*

David Bohm first brought the quantum potential to prominence in 1952. As yet it has never been observed directly or indirectly. We are running two experiments looking for the effects of the potential. First, in diffraction of argon atoms in the excited state Ar* 3P

_{2}. The second, using the Stern-Gerlach (SG) effect applied to the excited state of helium He* 2

^{3}S

_{1}. Principally we apply the STIRAP process in both experiments to observe the transverse velocity of the flow of the atoms on exiting both the diffraction grating and the SG magnet. I will explain the STIRAP process and show how we are going to exploit it in this endeavour.

**Gisin, Nicolas**

*Indeterminism is physical *

Deterministic chaos requires that the initial condition contains all the random information about the chaotic future evolution of the dynamical system. Accordingly:

- The initial conditions contain infinite information,
- God played all dice at the big-bang and coded all outcomes in the initial conditions.

Admittedly, events that just come out of the blue is not an appealing alternative.

Here, we assume some degree of indeterminacy: states of classical systems are not fully determined. Equivalently, real numbers are never fully determined: at each time, only computable approximations exist. As time passes, new information gets created which reduces the indeterminacy and improves the real number approximations as in intuitionistic mathematics. Accordingly, if the dynamical system is chaotic, its evolution is indeterministic.

**Hance, Jonte**

*Supermeasured: Violating Statistical Independence without violating statistical independence*

Bell’s theorem is often said to imply that quantum mechanics violates local causality, and that local causality cannot be restored with a hidden-variables theory. This however is only correct if the hidden-variables theory fulfils an assumption called Statistical Independence. Violations of Statistical Independence are commonly interpreted as correlations between the measurement settings and the hidden variables (which determine the measurement outcomes). Such correlations have been discarded as “fine-tuning” or a “conspiracy”. We here point out that the common interpretation is at best physically ambiguous and at worst incorrect. The problem with the common interpretation is that Statistical Independence might be violated because of a non-trivial measure in state space, a possibility we propose to call “supermeasured”. We use Invariant Set Theory as an example of a supermeasured theory that violates the Statistical Independence assumption in Bell’s theorem without requiring correlations between hidden variables and measurement settings.

**Hossenfelder, Sabine**

*Why Superdeterminism?*

I will summarize why superdeterminism solves a number of problems, how it does that, and how we could possibly test whether it’s correct.

**Kastner, Ruth**

*What is Retrocausation in the Transactional Interpretation?*

In this talk I discuss the latest incarnation of the Transactional Interpretation. The fully relativistic form that I have developed in recent years, known as “RTI”, has two distinct types of evolution—one unitary (reversible) and the other non-unitary (irreversible). The latter non-unitary process constitutes ‘measurement’ in the quantum context. I consider in what sense the term ‘retrocausation’ applies to each of these processes, and propose an ontological picture in which spacetime can be viewed as an emergent construct.

**Kent, Adrian**

Quantum Reality from Asymptotic Measurements: possibilities and problems

I review the idea [Phys. Rev. A 96, 062121 (2017)] of understanding quantum reality via a postulated hypothetical asymptotic measurement of the electromagnetic field and related proposals, and discuss more recent developments.

**Kinney, David**

*Corroboration Conditions for Cyclic Bayesian Networks*

Since the 1980s, Bayesian Network models have been used to represent the causal structure of systems. Verma and Pearl (1988) prove that in an acyclic Bayesian Network, there is a one-to-one correspondence between sets of variables that stand in the d-separation relation to one another in a graphical structure and sets of variables that are conditionally independent of each other in a probability distribution that is Markov to that structure. This result provides clear corroboration conditions for acyclic Bayesian Network models. However, Neal (2000) shows that the same result does not hold for cyclic Bayesian Networks. This poses difficulties for the corroboration of theories that posit causal loops. Expanding on work by Clarke et al.(2014), I show that cyclic Bayesian Networks can be re-written as acyclic, Dynamic Bayesian Networks that are infinitely long in two directions. These models allow us to use to Verma and Pearl’s result to state probabilistic corroboration conditions for theories with causal loops. I discuss the implications of these results for superdeterminism.

**Palmer, Tim**

*Discretisation of the Bloch Sphere, Fractal Invariant Sets and Bell’s Theorem*

Max Planck famously introduced the notion of discretised packets of energy, quanta, thus kickstarting the development of our most successful theory of physics, replacing classical theories in which energy varies continuously. Despite its success, however, the concepts of reality and local causality are deeply problematic in quantum mechanics. Such problems may lie at the heart of why it has been so difficult to synthesise quantum and gravitational physics.

Motivated by these issues, we apply Planck’s discretisation insight again, but this time to the continuum of quantum mechanics’ state space – complex Hilbert Space. A particular discretisation is discussed – one which draws on number theoretic properties of trigonometric functions. This leads to a model of quantum physics which is necessarily superdeterministic in character, that is to say violates the Statistical Independence assumption in Bell’s Theorem. Because of this, the model does not need to invoke concepts of indefinite reality or nonlocality to explain the violation of Bell’s inequality.

**Papatryfonosa, Konstantinos**

*Hydrodynamic quantum analogs: Superradiant emission and tunnelling of bouncing drops*

A decade ago, in a series of ground-breaking experiments Yves Couder, Emmanuel Fort and co-workers discovered that millimetre-sized droplets may self-propel along the surface of a vibrating fluid bath, guided by their own ‘pilot’ wave field. The resulting particle-wave association, exhibits several features previously thought to be peculiar to the microscopic, quantum realm. These include single-particle diffraction from single and double slits, tunnelling, wave-like statistics in confined geometries, quantized orbits in a harmonic potential and in a rotating frame, orbital levels splitting, spin states, and more.

We here present our results on the statistical correlations of a bipartite hydrodynamic system, establishing a hydrodynamic analogue of quantum superradiance. In the first part, we present a numerical study using pairs of droplets that may tunnel between adjacent cavities. The droplets are confined in two subsystems, which are separated by an intervening cavity. The droplets have a preference of residing one cavity over the other, each of them forming a 2-level system. By changing the length of the intervening cavity, we can precisely control the coupling between the two two-level systems. In the second part, we show an experimental analogue of the same quantum effect, but in a different hydrodynamic scenario. Static hydrodynamic Bell tests based on these settings are discussed.

**Price, Huw**

*Can two zigs make a zig-zag? Causal engineering on the Parisian model *

Costa de Beauregard (1953) discovered the Parisian zig-zag. He pointed out, initially as an objection to the EPR argument’s Locality assumption, that retrocausality might allow spacelike causality in an EPR experiment, without action at a distance. Given retrocausality, a causal influence might follow a zig-zag path, via the past lightcones of the two particles involved. But how would the zig-zag actually work? On the face of it, retrocausality only gets us half way, in either direction. It gives us two zigs – two retrocausal arrows, meeting at the source event in the past. How do we get from z*ig-zig* to z*ig-zag?* This talk, based on joint work with Ken Wharton, proposes an answer. In causal modelling terms, the two zigs give us a *collider* at the source. It is well known that colliders often produce causal artifacts; this is the problem of ‘collider bias’. But in special circumstances a collider can transmit genuine causal influence, and this may be the key to building a zig-zag.

**Reilly, Michele (presenter) and Lloyd, Seth**

*Closed timelike curves, game theory, and quantum foundations*

This talk discusses the implications of projective closed timelike curves (P-CTCs) for decision making. We show that P-CTCs solve the social dilemma problem in multiparty games: no sub-optimal Nash equilibria can occur if all players know each others’ future actions. We show that this advantage persists for experimentally realizable P-CTCs. We discuss the consequences of closed timelike curves for quantum foundations.

**Sen, Indrajit**

*Modelling relativistic effects on entangled systems using retrocausality *

The possibility of using retrocausality to obtain a fundamentally relativistic account of the Bell correlations has gained increasing recognition in recent years. It is not known, however, the extent to which these models can make use of their relativistic properties to account for relativistic effects on entangled systems. I will discuss a hypothetical relativistic Bell experiment, where one of the wings experiences time-dilation effects. I will show that the retrocausal Brans model (Found. Phys., 49(2), 2019) can be easily generalized to analyze this experiment, and that it gives an experimentally testable prediction (Proc. R. Soc. A*. *476(2234), 2020). I will argue that it is not clear at present, due to technical difficulties, if this prediction is reproduced by quantum field theory. I will conclude by discussing the experiment from the perspective of other hidden-variable models. The discussion will highlight an interesting relationship between the quantum state and particle distributions for retrocausal models with particle trajectories.

**Sutherland, Roderick**

*Marriage of Convenience*

In considering the subtleties associated with the possible existence of retrocausality, it will likely be helpful to have viable examples of retrocausal models on hand in order to gain some insight into the good and not-so-good features that can typically arise. A sample model will be presented here which hopefully can serve this role. It is formed by a convenient merging of two well-known models, namely the de Broglie-Bohm approach and the weak value

formalism, in order to incorporate the better features of each. In particular the resulting model is local, time-symmetric and Lorentz covariant, with no need for a preferred frame. Coming from the weak value side, the model encompasses all observables (i.e., it is not just a toy model for e.g., spin) with the ontology all residing in three dimensions (rather than configuration space) and with all conservation laws satisfied. From the de Broglie-Bohm side, the model provides a resolution of the measurement problem and also provides the same derivation of the Born rule as originally formulated by Bohm. The merger of the two models

is achieved by deriving it all from a single Lagrangian. The intention is for this framework to draw out the more unfamiliar characteristics likely to be encountered with retrocausal schemes and make them more evident.

**Vervoort, Louis**

*Probability Theory as a Physical Theory Points to Superdeterminism. *

We try to make superdeterminism more intuitive, notably by simulating a deterministic model system, a billiard game. In this system an initial ‘bang’ correlates all events, just as in the superdeterministic universe. We introduce the notions of ‘strong’ and ‘soft’ superdeterminism, in order to clarify debates in the literature. Based on the analogy with billiards, we show that superdeterministic correlations may exist as a matter of principle, but be undetectable for all practical purposes. This allows us to counter classical objections to superdeterminism, such as the claim that it would be at odds with the construction of new theories. Our main argument in favor of superdeterminism comes from probability theory. Probability theory as a physical theory is, in a sense, the most general physics theory available, more encompassing than relativity theory and quantum mechanics, which comply with probability theory. We show that superdeterminism has a greater explanatory power than its competitors: it can coherently answer three questions from probability theory for which other positions remain powerless (cf. L. Vervoort, *Entropy* **2019**, *21*(9), 848).

**Weinstein, Steven**

*PR boxes as a limiting case of the Boltzmann Machine model*

A conditional restricted Boltzmann machine (cRBM) is a type of machine learning model which can be taught to “learn” the EPR correlations, and more generally the correlations associated with any discrete quantum system, with arbitrary initial state and final measurement. This provides the archetype for a hidden variable theory in which the “hidden” units in the Boltzmann machine are the “hidden variables”. Boltzmann machines have their inspiration in classical statistical physics, and have a free parameter which corresponds to kT in an Ising-like model. Letting this parameter go to infinity gives rise to completely random behavior in the model, while letting it go to zero gives rise to extreme nonlocality, including PR boxes. In this brief talk I will show how this works, and consider what this might be telling us about the physics behind the cRBM model.

**Wharton, Ken**

*All Useful Superdeterminstic Models are Retrocausal*

Using the causal model framework popularized by Judea Pearl, one can mathematically distinguish superdeterministic models from retrocausal models. (See arXiv:1208.4119, 1906.04313.) To be useful, all superdeterminstic models must be supplemented with future inputs (measurement settings), which transforms them into retrocausal models (i.e. models with future-input-dependence). Attempting a “past-inputs-only” analysis on such models is shown to not only lose all meaningful aspects of causation, but also makes them incapable of prediction. These conclusions will be supported using analogous classical scenarios, such as retarded/advanced potentials in electromagnetism. The implication is that superdeterminists must not be averse to retrocausal models.