Canad.MaTh. Bull. vol. þ (Ë), z§Ë@ pp. Ëzh–Ëhþ hTTp://dx.doi.org/˧. Ëþh/CMB-z§Ëþ-§@@- ©Canadian MaThemaTical SocieTy z§Ëþ Discrete Space-time and Lorentz Transformations Gerd Jensen and ChrisTian Pommerenke AbsTracT. Alfred Schild esTablished condiTions where LorenTz TransformaTions map world-vecTors (cT, x, y, z) wiTh inTeger coordinaTes onTo vecTors of The same kind. _e problem was dealT wiTh in The conTexT of Tensor and spinor calculus. Due To Schild’s number-TheoreTic arguments, The subjecT is also inTeresTing when isolaTed from its physical background. Schild’s paper is noT easy To undersTand. _erefore, we ûrsT presenT a sTreamlined version of his proof which is based on The use of null vecTors. _en we presenT a purely algebraic proof ThaT is somewhaT shorTer. BoTh proofs rely on The properTies of Gaussian inTegers. 1 Introduction _e points of The Minkowski space-Time and The LorenTz TransformaTions are usually ⊺ ′ wriTTen in The form X⃗ = (T, x, y, z) ∈ R and X⃗ = LX⃗, where L is a real × maTrix ′ ′ ′ ′ and T z − x z − y z − z z = Tz − xz − yz − zz. _e speed of lighT is here assumed To be Ë. An equivalenT represenTaTion is obTained by The vecTor space ⎧ ⎫ ⎪ T + z x + iy ⎪ (Ë.Ë) M ∶= ⎨ X = ∶ T, x, y, z ∈ R⎬ ⎪ x − iy T − z ⎪ ⎩⎪ ⎭⎪ of HermiTian maTrices X. _ey saTisfy deT X = Tz −xz −yz −zz. Given such a HermiTian ⊺ maTrix X, wriTe X⃗ for The associaTed LorenTz vecTor (T, x, y, z)⊺ .If ⎧ ⎫ ⎪ a b ⎪ (Ë.z) Γ ∶= ⎨ A = ∶ a, b, c, d ∈ C, S deT AS = Ë⎬ ⎪ c d ⎪ ⎩⎪ ⎭⎪ ′ ∗ and A ∈ Γ, Then X ∶= AXA is again HermiTian and has a form analogous To X in (Ë.Ë) ′ ′ ′ ′ ′ wiTh uniquely deTermined T , x , y , z . Because deT X = deT X, The linear mapping ′ ∗ (Ë.h) X = AXA is a LorenTz TransformaTion. IT uniquely deTermines a real maTrix such ThaT × LA (Ë. ) ′ . X⃗ = LAX⃗ In [6], A. Schild found condiTions ThaT all components of are inTegers; an equiv- LA alenT formulaTion is ThaT maps The laTTice onTo itself. His main resulT is The LA Z following Theorem. Received by The ediTors January Ë, z§Ë . Published elecTronically OcTober zË, z§Ëþ. AMS subjecT classiûcaTion: zzE h, z§H,ÇhA§þ. Keywords: LorenTz TransformaTion, inTeger laTTice, Gaussian inTegers. Ëzh Downloaded from https://www.cambridge.org/core. 27 Sep 2021 at 22:51:20, subject to the Cambridge Core terms of use. Ëz G. Jensen and C. Pommerenke _eorem Ë.Ë (Schild) (i) LeT A ∈ Γ, deT A ∈ {Ë, i, −Ë, −i}, and eiTher (a) a, b, c, d ∈ Z + iZ and SaSz + SbSz + ScSz + SdSz even or (b) a, b, c, d i Ë Ë i . ∈ Z + Z + z ( + ) (Ë. ) _en LA deûned by has inTeger components. (ii) Γ Ë For every B ∈ such ThaTThe components of LB are inTegers, There is u ∈ C, SuS = such ThaT A ∶= uB fulûlls The condiTions in (i). If η ∈ {Ë, i, −Ë, −i}, and A saTisûes The condiTions in (i), Then so does ηA. Because deT(ηA) = ηz deT A = ± deT A, one can assume ThaT deT(ηA) = Ë or deT(ηA) = i. _erefore The conclusion deT A Ë, i, Ë, i in (ii) can be replaced wiTh deT A Ë, i . ∈ { − − } ∈ { √} _is is no loss of generaliTy for mulTiplicaTion wiTh a suiTable power of (Ë + i)~ z converts A ∈ Γ inTo an elemenT of SL(z, C). For Schild The physical aspects were essenTial, so The problem was dealT wiTh in The conTexT of Tensor and spinor calculus. However, mainly because of Schild’s num- ber TheoreTic arguments, The subjecT is also inTeresTing when isolaTed from its physical background. _erefore, a sTreamlined presenTaTion, which is reduced To The quinTes- senTial maThemaTical feaTures, mighT be useful. In The following, Schild’s proof is presenTed using only simple maTrix calculus. His meThod is geomeTrical in naTure;The problem is TranslaTed To mapping properTies of The spin TransformaTions A. AddiTionally, a more direcT, algebraic proof is given. IT musT be emphasized, however, ThaTSchild’s meThod points To aspects and problems ThaT could hardly be found in The course of a diòerenT reasoning. Some of These, e.g., The quesTion of The equivalence of Timelike vecTors wiTh inTeger components under LorenTz TransformaTions wiTh inTeger components, are dealT wiTh in [6]; furTher, There are ouTlined unpublished results of H. S. M. CoxeTer concerning The generaTors of The discreTe group; see also [ ]. BoTh proofs resT upon The unique facTorizaTion in The ring of Gaussian inTegers. _e crucial sTep in revealing The sTrucTure of The TransformaTion maTrices is in each case The comparison of The facTorizaTions of a number ThaT is boTh a square and a producT of Two diòerenT numbers (in The proofs of _eorem h. and Lemma .h). ATThe outseT, inTegral LorenTz maTrices are deûned as Those elements of Z × ThaT saTisfy The ˧ orThogonaliTy condiTions for The columns. _eorem Ë.Ë disenTangles The complexiTy signiûcanTly and brings iT down To Ç inTeger parameTers and z consTraints; whaT is more, The smaller seT of parameTers allows deeper insighT inTo The group sTruc- Ture. AnoTher inTeresTing and expliciT meThod was presenTed by J. D. Louck in [þ], based on The biquaTernionic parameTrizaTion of SL(z, C). 2 Preliminaries 2.1 Four-vectors _e connecTion beTween (Ë.h) and (Ë. ) can be made expliciT by means of The Kro- necker producT and The vecTorizaTion of maTrices ([h,Chap. ]). For z × z maTrices A aËË aËz and B, The Kronecker producT A B is The block maTrix aËË B aËz B , = azË azz ⊗ × azË B azz B and vec is The vecTor , , , ⊺ obTained by sTacking The columns of on A (aËË azË aËz azz) A Downloaded from https://www.cambridge.org/core. 27 Sep 2021 at 22:51:20, subject to the Cambridge Core terms of use. DiscreTe Space-Time and LorenTz TransformaTions Ëzþ Top of one anoTher.If C, D, X are also z × z maTrices, Then ⊺ (z.Ë) vec(AXB ) = (B ⊗ A) vec X, (z.z) (A ⊗ B)(C ⊗ D) = AC ⊗ BD, z.h) deT(A ⊗ B) = (deT A)z(deT B)z.( _e points of space-Time have The equivalenT represenTaTions X, X⃗, and vec X = ⊺ (T + z, x − iy, x + iy, T − z)⊺ . _e vecTors X⃗ and vec X are connecTed by a uniTary maTrix T: Ë §§ Ë vec Ë ⎛§ Ë i § ⎞ X ⎜ − ⎟ (z. ) √ = TX⃗ wiTh T ∶= √ ⎜ ⎟ . z z ⎜§ Ë i § ⎟ ⎝Ë §§ −Ë⎠ Using (z.Ë), relaTion (Ë.h) becomes ′ vec X = (A ⊗ A) vec X, and wiTh (z. ), iT follows ThaTThe maTrix in (Ë. ) has The represenTaTion LA (z.þ) ∗ . LA = T (A ⊗ A)T From (z.þ), (z.h), and (Ë.z) one obTains (z.@) deT Ë for Γ. LA = A ∈ WiTh (z.z) iT follows from (z.þ) ThaT , −Ë . LBA = LB LA LA = LA−Ë By subsTiTuTing special maTrices X in (Ë.h), one easily shows The following lemma. Ë Lemma z.Ë LA = LB holds if and only if B = uA wiTh u ∈ C, SuS = . From (z.þ) (in its expanded form ( .h)) and (z.@) iT is seen ThaT belongs To The LA group SO+ Ë, h of resTricTed LorenTz TransformaTions. AcTually The mapping ( ) A → LA + produces a double cover of SO (Ë, h) by SL(z, C) (see, for insTance, [Ë, Sec. @.h]), + so _eorem Ë.Ë characTerizes all members of SO (Ë, h) wiTh inTeger components. For Technical reasons, following Schild, we deal wiTh Γ insTead of SL(z, C), because This allows for a smaller number of case disTincTions. Moreover, mulTiplicaTion wiTh a suiT- √ able power of (Ë + i)~ z converts A ∈ Γ inTo an elemenT of SL(z, C). 2.2 Gaussian Integers LeT Z[i] = Z + iZ denoTe The ring of Gaussian inTegers and E = {Ë, i, −Ë, −i} its group of units. _e associaTes of x ∈ Z[i] are x, ix, −x, and −ix. _e prime elements of Z[i] are called Gaussian primes. _e Gaussian inTegers form a unique facTorizaTion domain ([z, Sec.Ëz.Ç]), which will be simply referred To by UFD;Two facTorizaTions of an inTeger are associaTes of each oTher. A naTural number ThaT is prime in Z is a Gaussian prime if and only if iT is congruenTTo −Ë (mod ) (see [z, Sec.Ëþ.Ë]); oTherwise, iT is The producT of Two complex conjugaTe Gaussian primes. If The greaTesT common divisor of real inTegers r, s, . is deTermined in Z and in Z[i], Downloaded from https://www.cambridge.org/core. 27 Sep 2021 at 22:51:20, subject to the Cambridge Core terms of use. Ëz@ G. Jensen and C. Pommerenke The results are associaTes; The posiTive value will be denoTed by gcd(r, s, ... ). For X as in (Ë.Ë) wiTh T, x, y, z ∈ Z, leT (z.6) δ(X) ∶= gcd(T, x, y, z). Because z = (Ë+i)(Ë−i), all real Gaussian primes are odd. For w ∈ Z[i] The producT ρ(w) of its posiTive Gaussian prime facTors is odd and has a facTorizaTion (z.Ç) ρ(w) = σ(w)r(w)z wiTh square-free σ(w). All facTors of ρ(w) are congruenTTo −Ë (mod ). For a ∈ Z[i] The pariTy π(a) ∈ {§, Ë} is deûned by π(a) ≡ Re a + Im a (mod z). If π(a) = §, Then a is called even; if π(a) = Ë, Then a is called odd. _aT exTends The Terminology for Z, for Im a = § if a ∈ Z, hence π(a) = § for even and π(a) = Ë for odd . FurThermore, is a ring homomorphism i . _e seT a π Z[ ] → Zz (z.) i z , ⊺ Z[ ]§ ∶= {(ξ§ ξË) ∶ π(ξ§) = π(ξË)} is a submodule of The Z[i]-module Z[i]z.IT conTains The vecTors Ë i § Ë (z.˧) + , , ; e§ ∶= eË ∶= ez ∶= § Ë + i Ë , is a basis of i z.