A temporal counterpart to Bell nonlocality would intuitively refer to the presence of non-classical correlations between timelike-separated events. The hypothesis of temporal nonlocality has received recent support in the literature, and its existence would likely influence the future development of physical theories. This paper shows how Adlam's principle of temporal locality can be violated within a protocol involving indefinite causal orders. While the derivations of Leggett-Garg inequalities or the temporal CHSH inequality are said to involve problematic assumptions preventing (...) a targeted probing of a well-defined notion of temporal nonlocality, the present test is free from such worries. However, it is shown that the test, in its current formulation, fails to be fully model-independent. We provide several considerations regarding the physicality of ICOs that could help alleviate this drawback. In the present work, a specific physical interpretation of ICOs in terms of retrocausal influences would explain the presence of temporally nonlocal correlations. It is argued that, as the physical underpinnings of temporal nonlocality might also account for standard Bell nonlocality, focusing on the former as a consequence of ICOs might support under-explored strategies to make sense of the latter. (shrink)

It is a frequent assumption that—via superluminal information transfers—superluminal signals capable of enabling communication are necessarily exchanged in any quantum theory that posits hidden superluminal influences. However, does the presence of hidden superluminal influences automatically imply superluminal signalling and communication? The non-signalling theorem mediates the apparent conflict between quantum mechanics and the theory of special relativity. However, as a ‘no-go’ theorem there exist two opposing interpretations of the non-signalling constraint: foundational and operational. Concerning Bell’s theorem, we argue that Bell employed (...) both interpretations, and that he finally adopted the operational position which is associated often with ontological quantum theory, e.g., de Broglie–Bohm theory. This position we refer to as “effective non-signalling”. By contrast, associated with orthodox quantum mechanics is the foundational position referred to here as “axiomatic non-signalling”. In search of a decisive communication-theoretic criterion for differentiating between “axiomatic” and “effective” non-signalling, we employ the operational framework offered by Shannon’s mathematical theory of communication, whereby we distinguish between Shannon signals and non-Shannon signals. We find that an effective non-signalling theorem represents two sub-theorems: Non-transfer-control theorem, and Non-signification-control theorem. Employing NTC and NSC theorems, we report that effective, instead of axiomatic, non-signalling is entirely sufficient for prohibiting nonlocal communication. Effective non-signalling prevents the instantaneous, i.e., superluminal, transfer of message-encoded information through the controlled use—by a sender-receiver pair —of informationally-correlated detection events, e.g., in EPR-type experiments. An effective non-signalling theorem allows for nonlocal quantum information transfer yet—at the same time—effectively denies superluminal signalling and communication. (shrink)

Nonlocality in quantum mechanics does not follow from nonseparability, nor does classical stochastic independence imply physical independence. In this paper an explicit proof of a Bell inequality is recalled, and an analysis of the Aspect experiment in terms of noncontextual, but indefinite weights, or improper probabilities, is given.

The nonlocality phenomena exhibited by entangled quantum systems are certainly one of the most extraordinary aspects of quantum theory. This book discusses this phe nomenon according to several points of view, i.e., according to different interpretations of the mathematics of the quantum formalism. The several interpretations of the Copenhagen interpretation, the many worlds, the de Broglie-Bohm, quantum logics, the decohering by the environment approach and the histories approach interpretations are scrutinized and criticized in detail. Recent results on cryptography, quantum (...) bit commitment, quantum erasers and teleportation are also presented and discussed. In preparing the book we benefited from discussions with many people, but we would like, in particular, to express our gratitude to Professor B. d'Espagnat for his useful comments and suggestions. We are grateful also to Ms. L. Gentry EI-Dash for the English revision, to Dr. 1. E. Maiorino for the production of the figures and a careful reading of the manuscript, and for the statI of Plenum for advice and for having produced a nice book. Finally, the authors thank FAPESP (contract no. I 99612657-0) for a grant making this book possible. A. A. ORIB AND W. A. RODRIGUES, JR. (shrink)

According to the current epistemic interpretation of quantum probabilities, the quantum correlations manifesting nonlocality can be derived from purely probabilistic and information-theoretic constraints. As such, they do not constitute a spacetime phenomenon and cannot lead to conflict between QM and any spatial-temporal constraints. This paper compares recent epistemic interpretations with earlier probabilistic interpretations, noting their merits as well as the difficulties they encounter. In particular, the implications of the recent PBR theorem are examined. While generally seen as undermining the (...) epistemic interpretation, I argue that the PBR theorem actually suggests an epistemic position more radical than previously ones. This review of probabilistic interpretations enables reassessment of the epistemic theorist's attempt to disentangle nonlocality from spacetime considerations. (shrink)

At first sight the Aharonov- Bohm effect appears nonlocal, though not in the way EPR/Bell correlations are generally acknowledged to be nonlocal. This paper applies an analysis of nonlocality to the Aharonov- Bohm effect to show that its peculiarities may be blamed either on a failure of a principle of local action or on a failure of a principle of separability. Different interpretations of quantum mechanics disagree on how blame should be allocated. The parallel between the Aharonov- Bohm effect (...) and violations of Bell inequalities turns out to be so close that a balanced assessment of the nature and significance of quantum nonlocality requires a detailed study of both effects. (shrink)

Nonlocality is a puzzling issue in modern physics. I propose that, aside from the experimental determination of nonlocality, the concept of atomistic lightmdash;discrete, self-bounded photonsmdash;breaks down toward something like nonlocality when subjected to philosophical scrutiny. Louis de Broglie made a similar argument regarding the material atom: the concept of the classical atom, when interrogated, collapses upon itself to offer a glimpse of wave-particle duality. Light atoms or photons, I argue, similarly collapse toward the contradictory possibility of (...) class='Hi'>nonlocality. (shrink)

give a proof of the existence of nonlocal influences acting on correlated spin-1/2 particles in the singlet state which does not require any particular interpretation of quantum mechanics (QM). (Except Stapp holds that the proof fails under a many-worlds interpretation of QM—a claim we analyse in 1.2.) Recently, in responding to Redhead's ([1987], pp. 90-6) criticism that the Stapp 1 proof fails under an indeterministic interpretation of QM, Stapp [1989] (henceforth Stapp 2), has revised the logical structure of his proof (...) including its crucial locality assumption. Our main aim is to show that this revision is a step in the wrong direction because it faces two difficulties which undermine the resulting proof's significance (3.1) and validity (3. 2). We also clarify and extend the Stapp 1 proof (1. 1) with the aid of Lewis' analysis of counterfactuals (1. 2) and causal dependence (2. 2 and 2. 3). In so doing, we are able to identify two new defects in the Stapp 1 proof (1. 3 and 2. 1) in addition to corroborating Redhead's criticism (2. 2). Also, the additional assumptions which save the Stapp 1 proof's validity are detailed (2. 3) and some new difficulties for the determinist are pointed out by exploiting a slightly extended version of the proof (2. 4). In providing this full analysis of the Stapp 1 proof, we also construct the necessary framework within which to provide a critique of Stapp 2's proof (3). *Portions of this paper were presented by R. K. Clifton to the 1988 British Society for the Philosophy of Science Conference at the University of Southampton. R. K. Clifton wishes to thank the Natural Sciences and Engineering Research Council of Canada, the Royal Commission for the Exhibition of 1851, and the Governing Body of Peterhouse at Cambridge University for support during this work. (shrink)

This paper reviews the origin of inertia according to Mach's principle and Weber's law of gravitation. The resulting theory is based on simultaneous nonlocal gravitational interactions between particles in the solar system and others in the remote universe beyond the Milky Way galaxy. It explains the precession of the perihelion of Mercury. A most important implication of the Mach-Weber theory of the force of inertia is the necessity for a large amount of uniformly distributed matter in the galactic universe. This (...) matter could be the source of the cosmic background radiation. Nonlocal inertia forces are compatible with a static universe and also with an expanding universe but the latter would demand slow changes in the mass of particles and the gravitational constant. (shrink)

We discuss Einstein's ideas on the need for a theory that is both objective and local and also his suggestion for realizing such a theory through nonlinear field equations. We go on to analyze the nonlocality implied by the quantum theory, especially in terms of the experiment of Einstein, Podolsky, and Rosen. We then suggest an objective local field model along Einstein's lines, which might explain quantum nonlocality as a coordination of the properties of pulse-like solutions of the (...) nonlinear equations that would represent particles. Finally, we discuss the implications of our model for Bell's inequality. (shrink)

In a recent article under the above title (but without the question mark) Henry Stapp has presented arguments which lead him to conclude that under suitable conditions “the truth of a statement that refers only to phenomena confined to an earlier time” must “depend on which measurement an experimenter freely chooses to perform at a later time.” I suggest that this conclusion contains an essential ambiguity as regards the meaning of the expression “statement referring only to phenomena confined to an (...) earlier time,” when such a statement talks about the outcome of an experiment that was not actually performed. As a result Stapp’s argumentation does not imply that future choices can affect present facts. But it does provide a wonderful and instructively different opportunity to reexamine a central point of Bohr’s famous reply to Einstein, Podolsky, and Rosen. (shrink)

Aiming to unravel the mystery of quantum mechanics, this book is concerned with questions about action-at-a-distance, holism, and whether quantum mechanics gives a complete account of microphysical reality. With rigorous arguments and clear thinking, the author provides an introduction to the philosophy of physics.

EPR experiments demonstrate that standard quantum mechanics exhibits the property of nonlocality , the enforcement of correlations between separated parts of an entangled quantum systems across spacelike separations. Nonlocality will be clarified using the transactional interpretation of quantum mechanics, and the possibility of superluminal effects (e.g., faster-than-light communication) from nonlocality and non-linear quantum mechanics will be examined.

In this essay I examine various aspects of the nearcentury-long debate concerning the conceptualfoundations of quantum mechanics and the problems ithas posed for physicists and philosophers fromEinstein to the present. Most crucial here is theissue of realism and the question whether quantumtheory is compatible with any kind of realist orcausal-explanatory account which goes beyond theempirical-predictive data. This was Einstein's chiefconcern in the famous series of exchanges with NielsBohr when he refused to accept the truth orcompleteness of a doctrine (orthodox QM) (...) which ruledsuch questions to be strictly inadmissible. I discussthe later history of quantum-theoretical debate withparticular reference to the issue of nonlocality,i.e., the phenomenon of superluminal(faster-than-light) interaction betweenwidely-separated particles. Then I show how thestandard `Copenhagen' interpretation of QM hasinfluenced current anti-realist orontological-relativist approaches to philosophy ofscience. Indeed, there are clear signs that somephilosophers have retreated from a realist positionvery largely in response to just these problems. So itis important to ask exactly why – on what scientificor philosophical grounds – any preferred alternative(causal-realist) construal should have been ruled outas a matter of orthodox QM wisdom. Moreconstructively, my paper presents various arguments infavour of one such alternative, the `hidden-variables'theory developed since the early 1950s by David Bohmand consistently marginalised by proponents of theCopenhagen doctrine. (shrink)

The effect of the rotating-wave approximation (RWA) on the coupling between an atom and the electromagnetic field is studied in the dipole approximation. It is demonstrated that use of the RWA results in an explicitly nonlocal interaction.

J. S. Bell's classic 1966 review paper on the foundations of quantum mechanics led directly to the Bell nonlocality theorem. It is not widely appreciated that the review paper contained the basic ingredients needed for a nonlocality result which holds in certain situations where the Bell inequality is not violated. We present in this paper a systematic formulation and evaluation of an argument due to Stairs in 1983, which establishes a nonlocality result based on the Bell-Kochen-Specker “paradox” (...) in quantum mechanics. (shrink)

The discussion of whether quantum theory is compatible with the locality idea that no causal influence can act outside the forward light cone has recently been tightly linked to the corresponding question for a much simpler model theory that enjoys all of the pertinent properties of quantum theory but is much easier to fully comprehend. It is shown here that, contrary to recent claims, the model theory is incompatible with the locality idea mentioned above. The logical structure of the argument, (...) which is similar to the one employed for quantum theory, is discussed. (shrink)

A collaboration between distinguished physicists and philosophers of physics, this important anthology surveys the deep implications of Bell's nonlocality theorem.

Bell’s theorem is purported to demonstrate the impossibility of a local “hidden variable” theory underpinning quantum mechanics. It relies on the well-known assumption of ‘locality’, and also on a little-examined assumption called ‘statistical independence’ (SI). Violations of this assumption have variously been thought to suggest “backward causation”, a “conspiracy” on the part of nature, or the denial of “free will”. It will be shown here that these are spurious worries, and that denial of SI simply implies nonlocal correlation between spacelike (...) degrees of freedom. Lorentz-invariant theories in which SI does not hold are easily constructed: two are exhibited here. It is conjectured, on this basis, that quantum-mechanical phenomena may be modeled by a local theory after all. (shrink)

Nonlocal reaction–diffusion equations describe various biological and biomedical applications. Their mathematical properties are essentially different in comparison with the local equations, and this difference can lead to important biological implications. This review will present the state of the art in the investigation of nonlocal reaction–diffusion models in biomedical applications. We will consider various models arising in mathematical immunology, neuroscience, cancer modelling, and we will discuss their mathematical properties, nonlinear dynamics, resulting spatiotemporal patterns and biological significance.

Nonlocal “realistic” Leggett models can be considered refuted by the before-before experiment. “Single preferred frame” models are not refuted by this experiment but bear severe oddities.

In this paper we present a series of computer calculations carried out in order to demonstrate exactly how the de Broglie-Bohm interpretation works for two-particle quantum mechanics. In particular, we show how the de Broglie-Bohm interpretation can account for the essential features of nonrelativistic, two-particle quantum mechanics in terms of well-defined, correlated, individual particle trajectories and spin vectors. We demonstrate exactly how both quantum statistics and the correlations observed in Einstein-Podolsky-Rosen experiments can be explained in terms of nonlocal quantum potentials (...) and nonlocal quantum torques which act on the well-defined individual particle coordinates and spin vectors. (shrink)

In the conventional approach to quantum mechanics, indeterminism is an axiom and nonlocality is a theorem. We consider inverting the logical order, making nonlocality an axiom and indeterminism a theorem. Nonlocal “superquantum” correlations, preserving relativistic causality, can violate the CHSH inequality more strongly than any quantum correlations.

We consider an integrable, nonlocal and nonlinear, Schrödinger equation as a model for building space–time patchings in inhomogeneous loop quantum cosmology. We briefly review exact solutions of the NNSE, specially those obtained through “geometric equivalence” methods. Furthemore, we argue that the integrability of the NNSE could be linked to consistency conditions derived from LQC, under the assumption that the patchwork dynamics behaves as an integrable many-body system.

We show that local realism applied to states characterized by a single quantum equally and coherently shared between a number of qubits (so-called W states) produces predictions incompatible with quantum theory. The origin of this incompatibility is shown to originate from the destructive interference of amplitude probabilities associated with nonlocal states, a phenomenon that has no classical analog.

According to D. Bohm’s interpretation of quantum mechanics, a particle always has a well-defined spatial trajectory. A change in boundary conditions can nonlocally change that trajectory. In this note we point out a striking instance of this phenomenon that is easy to understand qualitatively.

A new proof of the impossibility of reconciling realism and locality in quantum mechanics is given. Unlike proofs based on Bell's inequality, the present work makes minimal and transparent use of probability theory and proceeds by demonstrating a Kochen-Specker type of paradox based on the value assignments to the spin components of two spatially separated spin-1 systems in the singlet state of their total spin. An essential part of the argument is to distinguish carefully two commonly confused types of contextuality; (...) we call them ontological and environmental contextuality. These in turn are associated with two quite distinct senses of nonlocality. We indicate the relevance of our treatment to other related discussions in recent literature on the philosophy of quantum mechanics. (shrink)

In this paper, I argue that a satisfactory account of consciousness will involve both (1) local indeterminism, in that some relevant events are not wholly and unequivocally pre‐determined by immediately prior local events, and (2) nonlocality, in that, within the leeways left by local indeterminism, what happens can be immediately affected by spatially separated or extended events. I briefly consider if this can be avoided by treating consciousness as supervenient and epiphenomenal; and I suggest that current theories of consciousness (...) break down through failure to incorporate local indeterminism and nonlocality. I conclude by outlining possible implications for freedom of the will, physicalism, and the relevance of quantum mechanics to consciousness. (shrink)

In this paper, following an elementary line of thought which somewhat differs from the usual one, we prove once more that any deterministic theory predictively equivalent to quantum mechanics unavoidably exhibits a contextual character. The purpose of adopting this perspective is that of paving the way for a critical analysis of the use of counterfactual arguments when dealing with nonlocal physical processes.

Generalized Quantum Theory seeks to explain and predict quantum-like phenomena in areas usually outside the scope of quantum physics, such as biology and psychology. It draws on fundamental theories and uses the algebraic formalism of quantum theory that is used in the study of observable physical matter such as photons, electrons, etc. In contrast to quantum theory proper, GQT is a very generalized form that does not allow for the full application of formalism. For instance neither a commutator, such as (...) Planck’s constant, nor any additive operations are defined, which precludes the usage of a full Hilbert-space formalism. But it is a formalized phenomenological theory that is applicable whenever the core element of a quantum theory needs to be captured, namely in the presence of incompatible or non-commuting operations. As a consequence, it also predicts nonlocal, generalized entanglement correlations in systems other than proper quantum systems. In this paper we summarize the specific scientific evidence relating to the quantum-like mental, behavioral and physiological nonlocal correlations. Such non-local, generalized entanglement correlations are expected, both in space and time, between subsystems of a larger system, whenever observables pertaining to the global system are incompatible or complementary to observables pertaining to subsystems, as predicted by GQT. The result is a coherent explanation of a significant amount of controversial and seemingly weird occurrences that cannot be explained by classical physical laws. This review also offers a new perspective of the human mind’s potential. (shrink)

Quantum violation of Bell inequalities is now used in many quantum information applications and it is important to analyze it both quantitatively and conceptually. In the present paper, we analyze violation of multipartite Bell inequalities via the local probability model—the LqHV model, incorporating the LHV model only as a particular case and correctly reproducing the probabilistic description of every quantum correlation scenario, more generally, every nonsignaling scenario. The LqHV probability framework allows us to construct nonsignaling analogs of Bell inequalities and (...) to specify parameters quantifying violation of Bell inequalities—Bell’s nonlocality—in a general nonsignaling case. For quantum correlation scenarios on an N-qudit state, we evaluate these nonlocality parameters analytically in terms of dilation characteristics of an N-qudit state and also, numerically—in d and N. In view of our rigorous mathematical description of Bell’s nonlocality in a general nonsignaling case via the local probability model, we argue that violation of Bell inequalities in a quantum case is not due to violation of the Einstein–Podolsky–Rosen locality conjectured by Bell but due to the improper HV modelling of “quantum realism”. (shrink)

This essay aims to show that the recent development of quantum theory may provide us with an answer to one of the most compelling metaphysical problems, namely the problem of determinism. First, I sketch the conceptual background and draw the distinction between metaphysical and epistemological determinisms. Then, on the ground of the analysis of the problem of determinism in quantum mechanics, I argue that (1) metaphysical determinism is independent of quantum-mechanical formalism, and (2) that quantum nonlocality makes epistemological determinism (...) impossible. I also try to show that metaphysical determinism should be regarded as a Kantian regulative idea which sets the horizon for scientific inquiry but which at the same time lacks what Kant calls “objective validity.” The main conclusion is that metaphysical determinism should be regarded as an idealization – a formal, cognitive principle that we a priori assume and not something that we discover through scientific inquiry. (shrink)

It is assumed that the coupling of the field quantities Dμv (x) and F αβ (x) is nonlocal. This hypothesis leads to a theory of an electromagnetic field that has the following properties.(1) The source of the field F αβ (x) exhibits a center of charge and a center of mass that do not coincide, in general.(2) The field componentF 0i=−c2Ei is regular at the origin.(3) In the first-order approximation the new field equations are equivalent to the conventional Maxwell field (...) equations.(4) The conventional cutoff procedure in momentum space as practiced in the Maxwell-Lorentz theory is equivalent to the first-order approximation in terms of an invariant length ξ2.(5) The gyromagnetic ratio of the source of F αβ (x) is equal toc/mc for a quantum of chargee and massm. (shrink)

This book shines bright light into the dim recesses of quantum theory, where the mysteries of entanglement, nonlocality, and wave collapse have motivated some to conjure up multiple universes, and others to adopt a "shut up and calculate" mentality. After an extensive and accessible introduction to quantum mechanics and its history, the author turns attention to his transactional model. Using a quantum handshake between normal and time-reversed waves, this model provides a clear visual picture explaining the baffling experimental results (...) that flow daily from the quantum physics laboratories of the world. To demonstrate its powerful simplicity, the transactional model is applied to a collection of counter-intuitive experiments and conceptual problems. (shrink)

I derive a Gleason-type contradiction from assumptions weaker than those needed to reach a Bell inequality. By establishing the inconsistency between local realism and QM's perfect EPR-type anticorrelations, the proof fills in a gap left open by Bell arguments.

It is argued that among possible nonlocal hidden-variable theories a particular class (called here “crypto-nonlocal” or CN) is relatively plausible on physical grounds. CN theories have the property that (for example) the two photons emitted in an atomic cascade process are indistinguishable in their individual statistical properties from photons emitted singly, and that in the latter case the effects of nonlocality are unobservable. It is demonstrated that all CN theories are constrained by inequalities which are violated by the quantum-mechanical (...) predictions; these inequalities bear no simple relation to Bell's inequalities, and an explicit example is constructed of a CN theory which violates the latter. It is also shown that while existing experiments cannot rule out general CN theories, they do rule out (subject to a few caveats such as the usual ones concerning the well-known “loopholes”) the subclass in which the photon polarizations are linear. (shrink)

An interesting link between two very different physical aspects of quantum mechanics is revealed; these are the absence of third-order interference and Tsirelson’s bound for the nonlocal correlations. Considering multiple-slit experiments—not only the traditional configuration with two slits, but also configurations with three and more slits—Sorkin detected that third-order (and higher-order) interference is not possible in quantum mechanics. The EPR experiments show that quantum mechanics involves nonlocal correlations which are demonstrated in a violation of the Bell or CHSH inequality, but (...) are still limited by a bound discovered by Tsirelson. It now turns out that Tsirelson’s bound holds in a broad class of probabilistic theories provided that they rule out third-order interference. A major characteristic of this class is the existence of a reasonable calculus of conditional probability or, phrased more physically, of a reasonable model for the quantum measurement process. (shrink)

It is argued that the validity of the predictions of quantum theory in certain spincorrelation experiments entails a violation of Einstein's locality idea that no causal influence can act outside the forward light cone. First, two preliminary arguments suggesting such a violation are reviewed. They both depend, in intermediate stages, on the idea that the results of certain unperformed experiments are physically determinate. The second argument is entangled also with the problem of the meaning of “physical reality.” A new argument (...) having neither of these characteristics is constructed. It is based strictly on the orthodox ideas of Bohr and Heisenberg, and has no realistic elements, or other ingredients, that are alien to orthodox quantum thinking. (shrink)

The Bell 1964 theorem states that nonlocality is a necessary feature of hidden variable theories that reproduce the statistical predictions of quantum mechanics. In view of the no-go theorems for non-contextual hidden variable theories already existing up to 1964, and due to Gleason and Bell, one is forced to acknowledge the contextual character of the hidden variable theory which the Bell 1964 theorem refers to. Both the mathematical and the physical justifications of this contextualism are reconsidered. Consequently, the role (...) of contextualism in recent no-hidden-variables proofs and the import of these proofs are investigated. With reference to the physical intuition underlying contextualism, the possibility is considered whether a context-dependence of individual measurement results is compatible with context-independence of the statistics of measurement results. (shrink)

The status of a causal approach to EPR-Bell nonlocal correlations in terms of a counterfactual framework for causation is considered. It is argued that when the relativistic spacetime structure of the events is taken into due account, the adoption of this approach is best motivated by the assumption of a preferred frame of reference, an assumption that seems even more in need of justification than the causal theory itself.

We reconsider the problem of the compatibility of quantum nonlocality and the requests for a relativistically invariant theoretical scheme. We begin by discussing a recent important paper by T. Norsen on this problem and we enlarge our considerations to give a general picture of the conceptually relevant issue to which this paper is devoted.

We argue that certain types of many minds (and many worlds) interpretations of quantum mechanics, e.g. Lockwood ([1996a]), Deutsch ([1985]) do not provide a coherent interpretation of the quantum mechanical probabilistic algorithm. By contrast, in Albert and Loewer's ([1988]) version of the many minds interpretation, there is a coherent interpretation of the quantum mechanical probabilities. We consider Albert and Loewer's probability interpretation in the context of Bell-type and GHZ-type states and argue that it implies a certain (weak) form of (...) class='Hi'>nonlocality. 1 Introduction 2 Albert and Loewer's interpretation 3 Probabilities in Lockwood's interpretation 4 Sets of minds and their correlations 5 Many minds and GHZ. (shrink)