## Abstract

David Finkelstein was very fond of the new information-theoretic paradigm of physics advocated by John Archibald Wheeler and Richard Feynman. Only recently, however, the paradigm has concretely shown its full power, with the derivation of quantum theory (Chiribella et al., Phys. Rev. A **84**:012311, 2011; D’Ariano et al., 2017) and of free quantum field theory (D’Ariano and Perinotti, Phys. Rev. A **90**:062106, 2014; Bisio et al., Phys. Rev. A **88**:032301, 2013; Bisio et al., Ann. Phys. 354:244, 2015; Bisio et al., Ann. Phys. **368**:177, 2016) from informational principles. The paradigm has opened for the first time the possibility of avoiding physical primitives in the axioms of the physical theory, allowing a re-foundation of the whole physics over logically solid grounds. In addition to such methodological value, the new information-theoretic derivation of quantum field theory is particularly interesting for establishing a theoretical framework for quantum gravity, with the idea of obtaining gravity itself as emergent from the quantum information processing, as also suggested by the role played by information in the holographic principle (Susskind, J. Math. Phys. **36**:6377, 1995; Bousso, Rev. Mod. Phys. **74**:825, 2002). In this paper I review how free quantum field theory is derived without using mechanical primitives, including space-time, special relativity, Hamiltonians, and quantization rules. The theory is simply provided by the simplest quantum algorithm encompassing a countable set of quantum systems whose network of interactions satisfies the three following simple principles: homogeneity, locality, and isotropy. The inherent discrete nature of the informational derivation leads to an extension of quantum field theory in terms of a quantum cellular automata and quantum walks. A simple heuristic argument sets the scale to the Planck one, and the currently observed regime where discreteness is not visible is the so-called “relativistic regime” of small wavevectors, which holds for all energies ever tested (and even much larger), where the usual free quantum field theory is perfectly recovered. In the present quantum discrete theory Einstein relativity principle can be restated without using space-time in terms of invariance of the eigenvalue equation of the automaton/walk under change of representations. Distortions of the Poincaré group emerge at the Planck scale, whereas special relativity is perfectly recovered in the relativistic regime. Discreteness, on the other hand, has some plus compared to the continuum theory: 1) it contains it as a special regime; 2) it leads to some additional features with GR flavor: the existence of an upper bound for the particle mass (with physical interpretation as the Planck mass), and a global De Sitter invariance; 3) it provides its own physical standards for space, time, and mass within a purely mathematical adimensional context. The paper ends with the future perspectives of this project, and with an Appendix containing biographic notes about my friendship with David Finkelstein, to whom this paper is dedicated.

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## Notes

The problem of ordering is avoided miraculously thanks to the fortuitous non occurrence in nature of Hamiltonians with products of conjugated observables.

The two associations can be connected through the GNS construction.

More generally the map \(\mathcal {A}\) is an automorphism of the algebra.

Richard Feynman is reported to like the idea of finite information density, because he felt that:

*“There might be something wrong with the old concept of continuous functions. How could there possibly be an infinite amount of information in any finite volume?”*[61].If two transition matrices \(A_{h_{1}}=A_{h_{2}}\) are equal, we conventionally associate them with two different labels

*h*_{1}≠*h*_{2}in such a way that \({\sum }_{f\in N_{\pi (g)}}A_{\pi (g)f}\psi _{\pi ^{-1}(f)}={\sum }_{f\in N_{g}}A_{gf}\psi _{f}\). If such choice is not unique, we will pick an arbitrary one, since the homogeneity requirement implies that there exists a choice of labeling for which all the construction that will follow is consistent.The above arbitrariness is inherent the very notion of group presentation and corresponding Cayley graph, and will be expoited in the following, in particular in the definition of isotropy.

The absence of the appropriate mathematics was the reason of the lack of consideration of a discrete structure of space-time in earlier times. Einstein himself was considering this possibility and lamented such lack of mathematics. Here a passage reported by John Stachel [20]

*But you have correctly grasped the drawback that the continuum brings. If the molecular view of matter is the correct (appropriate) one, i. e., if a part of the universe is to be represented by a finite number of moving points, then the continuum of the present theory contains too great a manifold of possibilities. I also believe that this too great is responsible for the fact that our present means of description miscarry with the quantum theory. The problem seems to me how one can formulate statements about a discontinuum without calling upon a continuum (space-time) as an aid; the latter should be banned from the theory as a supplementary construction not justified by the essence of the problem, which corresponds to nothing “real”. But we still lack the mathematical structure unfortunately. How much have I already plagued myself in this way!*One should consider that the Dehn’s problem of establishing if two words of generators correspond to the same group element is generally undecidable. The same is true for the problem of establishing if the presentation corresponds to the trivial group!

The Brillouin zone is a compact subset of \(\mathbb {R}^{3}\) corresponding to the smallest region containing only inequivalent wave-vectors

**k**. (See Ref. [22] for the analytical expression.)The highest momentum observed is that of a ultra-high-energy cosmic ray, which is

*k*≪ 10^{−8}.A fast numerical technique to evaluate the quantum walk evolution numerically exploits the Fourier transform. For an application to the Dirac quantum walk see Ref. [31].

Also more generally one has

*A*=*T*_{ h }.The first Brillouin zone

*B*for the BCC lattice is defined in Cartesian coordinates as \(-{\sqrt 3}\pi \leq k_{i}\pm k_{j}\leq {\sqrt 3}\pi , \, i\neq j\in \{x,y,z\}\).This can also be e. g. the case of an overall phase independent of

**k**.Also the solutions with walk

*B*^{±}= (*A*_{ k })^{T}are contained in (22), since they can be achieved either by a shift in the Brillouin zone or as*σ*_{ y }*B*^{±}*σ*_{ y }=*A*^{±‡}, with the exchange of the upper and lower diagonal blocks that can be done unitarily.For

*d*= 1, modulo a permutation of the canonical basis, the quantum walk corresponds to two identical and decoupled*s*= 2 walks. Each of these quantum walks coincide with the one dimensional*Dirac walks*derived in Ref. [11]. The last one was derived as the simplest*s*= 2 homogeneous quantum cellular walk covariant with respect to the parity and the time-reversal transformation, which are less restrictive than isotropy that singles out the only Weyl quantum walk in one space dimension.Notice that, depending on the quantum walk

*A*^{+}(**k**) of*A*^{−}(**k**) in (16) we obtain corrections to the speed of light with opposite sign.Discreteness has doubled the particles: this corresponds to the well known phenomenon of Fermion doubling [73].

Very flattering are the compliments of Federico Faggin, the designer of the first microprocessor at Intel.

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## Acknowledgments

The present long-term unconventional project has needed a lot of energy and determination in the steps that had to be faced within the span of more than seven years. The work done up to now would have not been possible without the immeasurable contribution of some members of my group QUit in Pavia, as it can be seen from the list of references. All of them embraced with enthusiasm the difficult problems posed by the program, at the risk of their careers, in a authentically collaborative interaction. In particular, I am mostly grateful to Paolo Perinotti, with whom I had the most intense and interesting interactions of my entire career. I’m then very grateful to my postdocs Alessandro Bisio e Alessandro Tosini for their crucial extensive contribution, and to my PhD students Marco Erba and Nicola Mosco, and my previous PhD student Franco Manessi. I am very grateful to my long-date friend Matt Brin for introducing me to some among the top mathematicians in geometric group theory, which otherwise it would have been impossible for me to meet. In particular: Benson Farb, Dennis Calegari, Cornelia Drutu, Romain Tessera, and Roberto Frigerio. I personally learnt a lot from Benson Farb in four meetings in at the Burgeois Pig café in Chicago, during my august visits at NWU in Evanston, and am grateful to Dennis Calegari for two interesting meetings at UC. With Paolo Perinotti and Marco Erba we have visited Cornelia Drutu in Oxford, Romain Tessera in Paris, and Roberto Frigerio in Pisa, and from them we could learnt fast crucial mathematical notions and theorems, which otherwise it would have taken ages for us to find in books and articles. I want then to acknowledge some friends that enthusiastically supported me in the difficult stages of the advancement of this program, in particular my mentor and friend Attilio Rigamonti, and my friends Giorgio Goggi, Catalina Curceanu, Marco Genovese, all of them experimentalists, along with the theoreticians Lee Smolin, Rafael Sorkin, Olaf Dryer, Lucien Hardy, Kalamara Fotini Markopoulou, Bob Coecke, Tony Short, Vladimir Buzek, Renato Renner, Wolfgang Schleich, Lev B. Levitin, and Andrei Khrennikov, for appreciating the value of this research since the earlier heuristic stage. For inspiring scientific discussions I like to acknowledge Seth Lloyd, Reinhard F. Werner, Norman Margolus, Giovanni Amelino-Camelia, Shahn Majid, Louis H. Kauffman, and Carlo Rovelli, whereas I wish to thank Arkady Plotnitsky and Gregg Jaeger for very exciting discussions about history and philosophy of physics. I want finally to remark again the great help that I got from David Finkelstein, of whom I have been honored to be friend, and whose enthusiasm have literally boosted the second part of this project. Financially I acknowledge the support of the John Templeton foundation, whithout which the present project could had never take off from the preliminary heuristic stage.

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Work supported by the Templeton Foundation under the project ID# 43796 *A Quantum-Digital Universe*.

## Appendix

### Appendix

ᅟ

### Appendix A: A Very Brief Historical Account

The first very preliminary heuristic ideas about the current quantum cellular automaton/quantum walk theory have been presented in a friendly and open-minded environment at Perimeter Institute in Waterloo in a series of three talks in 2010-11 [35–37].^{Footnote 20} Originally, the idea of foliations over the quantum circuit has been explored, showing how the Lorentz time-dilations and space-contractions emerge by changing the foliation. This work has lead to the analyses of Alessandro Tosini in Refs. [51, 52] and in the conclusive work [54]. However, it was soon realized that the foliation on the quantum circuit explores only the causal connectivity of the automaton, and works in the same way also for a classical circuit, as it happens for random walks in one dimension (see e. g. Ref. [72]). Moreover, only rational boosts can be used, with the additional artifact that the events have to be coarse-grained in a boost-dependent way, with very different coarse-graining for very close values of the boost. This makes the recovery of the usual Lorentz transformations at large scales practically unfeasible. On the other hand, the first Dirac automaton in one space dimension [24] exhibited perfect Lorentz covariance for small wavevectors, which made clear that the quantum nature of the circuit plays a pivotal role in recovering the Lorentz invariance. In the same Ref. [24] it also emerged that the Dirac mass has to be upper bounded as a consequence of unitarity.

The idea that so-called “conventional”principles as homogeneity and isotropy may play a special role entered the scene since the very beginning [35] through the connection with the old works of Ignatowsky [80], whereas ideas about how to treat gauge theories emerged already in Ref. [37]. However, the project remained stuck for a couple of years because of two dead ends. First, we were looking to the realization of the quantum cellular automaton in terms of circuit gates, and we much later realized that the problem of connecting the gate realization (socalled Margolus scheme [77]) to the linear quantum walk was a highly non trivial problem for dimension greater than one. Second, we where considering Jordan-Wigner mappings between local qubits and discrete Fermions [53], generalizing to dimensions *d* > 1 what can be done for d=1, and later Tosini realized that for *d* > 1 such mapping cannot be done iso-locally [28, 29], namely preserving the locality of interactions. Paolo Perinotti, inspired from the work of Bialynicki-Birula [7], recognized the first Dirac quantum walks in 2 and 3 dimensions. Later the graph structure of the walk was pointed out to be a Cayley graph of a group by Matt Brin, and the work of the derivation from principles of Weyl and Dirac [22] followed after a Paolo’s nontrivial solution of the unitarity conditions. This was the turning point of the whole program. It was soon recognized that the Maxwell field could be obtained by tensor product of two Weyl, and Alessandro Bisio soon found a way of achieving the photons with entangled pairs of Fermions. We finally realized the pivotal role played by the eigenvalue equation of the quantum in restating the relativity principle and recovering Lorentz covariance, and Bisio found the construction recovering the notion of particle as invariant of the deformed Poincaré group.

### Appendix B: My Encounter with David Ritz Finkelstein

Vieque Island, January 6th 2014: FQXi IV International Conference on *The Physics of Information*. The conference is very interactive, mostly devoted to debates, round tables, and working groups. Max Tagmark organizes and chaires a morning session made of five-minutes talks. The audience includes distinguished scientists, a unique opportunity for presenting my Templeton project *A Quantum-Digital Universe*. I want to say many things that I consider very important, and I prepare my talk carefully, measuring the time of each single sentence, and memorizing each single word. The result goes beyond my best expectations, with gratifying comments by a number of scientists, some whom I meet for the first time.^{Footnote 21} But the best that happens is that a beautiful old man, whom I never met before, with a white bear and a hat, literally embraces me with a great smile, and almost with tears in his blue eyes says that I realized one of his dreams. His enthusiasm, so passionate and authentic captures me. I spend most of the following days discussing with him. He invites me to visit him in his home in Atlanta.

I visit David on March 16th and 17th in a weekend during a visit in Boston. His house is beautiful, with large windows opened on a surrounding forest. With his wife Shlomit we have pleasant conversations, some about their past encounter with the Dalai Lama.

David writes a nice dedication on my copy of his last book [58]. He then asks me to explain to him the derivation of quantum theory from information-theoretical principles (which I did with my former students Paolo Perinotti and Giulio Chiribella [18]: a textbook from Cambridge University Press is now in press [25]). I spend almost the two entire days in front of a small blackboard in David room full of books (see Fig. 8), drawing diagrams and answering to his many questions. His genuine interest will boost my enthusiasm for the years to come.s

After that visit David and I will continue to exchange emails. David regularly will send to me updates of his work. We promise to exchange visits soon, but unfortunately this will not happen again.

### Appendix C: My Talk at FQXi 2014 *Verbatim*

I’d like to tell you about the astonishing power of taking information more fundamental than matter, the informational paradigm advocated by Wheeler, Feynman, and Seth Lloyd of “the universe as a huge quantum computer”. Quantum Theory is indeed a theory of information, since it can be axiomatically derived from six axioms of pure information-theoretical nature. Five of the postulates are in common with classical information. The one that discriminates between quantum and classical is the principle of conservation of information–technically the purification postulate. Information means describing everything in terms of input-output relations between events/transformations, mathematically associating probabilities to closed circuits between preparations and observations. [Some of the principles are conceptually quite new and interesting, such as the local discriminability one, which in the quantum case reconciles holism with reductionism, with the possibility of achieving complete information by local observation.]

Now these postulates provide only the quantum theory of abstract systems, not the mechanical part of the theory. In order to get this you need to add new principles that lead to quantum field theory, without assuming relativity and space-time. These principles describe the topology of interactions, which determine the flow of information along the circuit. The first of these requirements is that of finite info-density, corresponding to having a numerable set of finite-dimensional quantum systems in interaction. Such principle, along with the assumption of unitarity, locality, homogeneity, isotropy and minimal dimension of the systems in interaction, are equivalent to minimizing the quantum algorithmic complexity of the information processing, reducing the physical law to a bunch of few quantum gates, and leading to a description in terms of a Quantum Cellular Automaton.

Now, it turns out that from these few assumptions only two quantum cellular automata follow that are connected by CPT, and Lorentz covariance is broken. They both converge to the Dirac equation in the relativistic limit of small masses and small wave-vectors. In the ultra-relativistic limit of large wave-vectors or masses (corresponding to a Planck scale) Lorentz covariance becomes only an approximate symmetry, and one has an energy scale and length scale that are invariant in addition to the speed of light, corresponding to the Doubly Special Relativity of Amelino-Camelia/Smolin/Magueijo, with the phenomenon of relative locality, namely that also coincidence in space, not only in time, is observer-dependent. The covariance is given by the group of transformation leaving the dispersion relations of the automaton invariant, and holds for energy-momentum. When you get back to space-time via Fourier, then you recover a space-time of quantum nature, with space-time points in superposition.

The quantum cellular automaton can be regarded as a theory unifying scales ranging from Planck to Fermi. It is interesting to notice that the same quantum cellular automaton also gives the Maxwell field, interestingly in the form of the de Broglie-Fermi neutrino theory of the photon. With the principle of bounded information density, also the Boson becomes an emergent notion, but the relation with Fermions is subtle in terms of localization. The fact that the theory is discrete avoids all problems that plague quantum field theory arising from the continuum, especially the problem of localization, but, most relevant, the theory is quantum *ab initio*, with no need of quantization rules. And this is the great bonus of taking information as more fundamental than matter.

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D’Ariano, G.M. Physics Without Physics.
*Int J Theor Phys* **56**, 97–128 (2017). https://doi.org/10.1007/s10773-016-3172-y

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DOI: https://doi.org/10.1007/s10773-016-3172-y

### Keywords

- Quantum fields axiomatics
- Quantum automata
- Walks
- Planck scale