Anomaly in sign function - probability function integration
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1 Anomly in sign function - probbility function integrtion Hn GEURDES Aress Geures Dtscience, KvK 6450, C v Lijnstrt 164, 593 NN, Den Hg Netherlns Abstrct E-mil :hn.geures@gmil.com In the pper it is emonstrte tht integrtion of proucts of sign functions n probbility ensity functions such s in Bell s formul for ±1 mesurement functions, les to inconsistencies. Keywors Inconsistency, Bell s theorem. 1 Introuction In 1964, John Bell wrote pper [1] on the possibility of hien vribles [] cusing the entnglement correltion E(, b between two prticles. In his fmous pper, Einstein [] rgue tht the quntum escription must be supplemente with extr vribles to explin the entnglement phenomenon. von Neumn [4] presente mthemticl proof tht ny hien vribles theory is in conflict with quntum mechnics. However, one cn oubt if von Neumn s view on the mtter ws completely relte to the physics [5]. An intereste reer cn fin more etil in preprint of Dieks [6]. In the present pper, n inconsistency in the strting formul of Bell [1] will be emonstrte. Bell bse his hien vrible escription on prticle pirs with entngle spin, originlly formulte by Bohm [3]. Bell use hien vribles λ tht re elements of universl set Λ n re istribute with ensity ρ(λ 0. Suppose, E(, b is the correltion between mesurements with istnt A n B tht hve unit-length, i.e. = b = 1, rel 3 im prmeter vectors n b. Then with the use of the λ we cn write own the clssicl probbility correltion between the two simultneously mesure spins of the prticles. This is wht we will cll Bell s formul. E(, b = ρ(λa(, λb( b, λλ (1.1 λ Λ The spin mesurement functions re, A(, λ { 1, 1} n B( b, λ { 1, 1}. The probbility ensity is normlize, ρ(λλ = 1. Exmple of inconsistency Bell s formul (1.1 is generl. Tht mens tht it hs to be vli for ll kins sub-cses where ±1 functions re use....
2 Geures In the present exmple we will concentrte the ttention on two ifferent expressions for e.g. A = A(, λ { 1, 1}. Let us lso mke use of one single Gussin ensity function n chnge the nottion slightly. We hve = 1 π x e y / y (.1 Hence, ρ Guss (x = in this exmple. To be more precise, we concentrte on sub-cse of Bell s formul in which e.g. E = A(, xρ Guss (x (. is prt of the computtion of more complete correltion. As n instructive exmple we my look t the cse where A(, λ = A (1 ( 1, λ 1 A ( (, λ A (3 ( 3, λ 3 = ±1. Ech A (m = ±1, with, m = 1,, 3. Similrly for B with B( b, λ = B (1 (b 1, λ 4 B ( (b, λ 5 B (3 (b 3, λ 6 = ±1 n six-fol norml ensity with {λ k } 6 k=1 inepenent vribles. The subsequently presente cses, lrey nticipte in (., refer to e.g. A (1 ( 1, λ 1. For ese of nottion represents 1 n x represents λ 1. Ech B (m = ±1, with, m = 1,, 3..1 Definition of to be use ±1 functions Let us efine, for, x R, A 1 (, x = 4H(x 1 H(x + 1 = +1, x 1, x < (.3 n A (, x = H(x + 1 4H(x 1 = +1, x 1, x < (.4 To remin close to the physics of the problem, we cn tke [ 1, 1]. In both expressions, H(x = 1 x 0 n H(x = 0 x < 0. The close limit Hevisie form is here equl to: H(x = lim ( exp e nx 0 (.5 n n It must be note tht (.3 n (.4 re perfectly in orer when in Bell formul we re looking for A { 1, 1}. Eqution (.5 is vli expression for the Hevisie function. No nee to bsolutely must hve H(0 = 1/.. A 1 form integrtion Let us compute with prtil integrtion E 1 = (.6 H + 1 Here n below, H, is n bbrevition of H(x. As fr s we know prtil integrtion is vli step in the concrete mthemtics behin Bell s correltion formul (1.1. It cn then be checke tht E 1 = H ( H + 1
3 Anomly. 3 = 1 + {( 4(H + 1} (H + 1 = = 1 6 = (H ( H The Gussin in (.1 gives P(+ = 1..3 A form integrtion H P( P(+ + P( = 1 P( Let us lso compute with prtil integrtion H + 1 E = (.8 It cn lso be checke tht E = H = 1 ( ( H + 1 {( 4(H + 1} = ( = 1 3 ( 1 (.9 = = = P( P( = 1 P( Of course, P( = 0. Despite the fct tht A = 1 A 1, the outcome of integrtion is the sme for A 1 s well s for A. We might hve expecte tht, bse on the erivtion: A (, x = (A 1(, x 1 1 = {A 1 (, x} A 1(, x = A 1(, x n {A 1 (, x} = 1, the integrls, E 1 in (.7 n E in (.9 woul iffer. So, where is the nomly? Note however, we my employ (.4 in the evlution of E. Let us, gin, look t the step E = (.10 (
4 4 Geures in the erivtion (.9. We my rewrite, noting H + 1 > 0, E = = = ( (.11 ( H + 1 In this form we cn recognize (.4 squre. So we lso my write = = 1 3 E = = 1 3 H ( H (H = P( + P(+ P( (.1 3 = 1 + P( Hence, the nomly with sign functions n probbility ensities, expecte from A(, x (A(, x 1 ( surfces when use is me of A = H(x +1 4H(x 1 = 1, in eqution (.11. Obviously, it is llowe to rewrite the integrl by multiplying the integrn with (H+1 (H+1 by efinition, either H + 1 = 1 or H + 1 = 3. Moreover, reshuffling of terms 1 ( = 1 ( = 1 ( = 1 = 1. Note lso tht, ( H + 1 is llowe. It enbles the use of (.4 squre, which is unity. So, it mkes perfect sense to hve 1 ( = 1 (.13 when H = 0 or H = 1, such s in (.5. The step from (.11 to (.1 is therefore justifie. The use of, e.g. prtil integrtion is llowe s prt of the rulebook of concrete mthemtics. If prtil integrtion must be exclue from the list of vli opertions, then we my sk if Bell s theorem is s generlly vli s is clime. In prticulr, looking t the step from (.11 to (.1, scepticl reer hs to emonstrte, observing (.13, wht is wrong with the step ( = H = n then the subsequent steps in (.1. Moreover, if looking t the bove eqution it is foun wnting, then how cn we be sure tht the ctivities leing to Bell inequlities (e.g. the CHSH inequlity re correct? A similr type of resoning, i.e. A = 1 in the integrn, is followe in the erivtion of the CHSH. For etile CHSH erivtion see [7].
5 Anomly. 5 3 Conclusion In the present pper support ws foun for the fct tht Bell s formul is the origin of inconsistent mthemtics. In this cse, noting A = 1 A 1, the nomly surfce without the A = 1 A prtil integrtion brekown in [7]. Nmely A(, x (A(, x 1 Nevertheless the nomly, lrey rgue for in [7], resurfce s cse of inconsistent outcomes where vli mthemticl opertions on the integrn re performe. We foun, E 1 E, uner certin group of vli opertions. The opertions on the integrn coincie with those employe in the CHSH erivtion. It ppers sfe to conclue tht the foun nomly in Bell s resoning [7], is not something ccientl. It is note tht the centrl role this theorem plys in fountion iscussions, oesn t put this (physics theorem beyon ny possible criticism. We showe, tht the form in which it shows the nomly is ifferent in ifferent cses. The nomly lso explins why it is possible to come with vli countermoel [7] n euce concrete mthemticl incompleteness. The ltter refers to the Göel phenomenon in concrete mthemtics [8]. It must lso be note tht the work of Noren [9] is relevnt here becuse the Bell theorem might not necessrily be centrl in the fountion iscussion. Obviously, the present mthemticl n fountionl iscussion is relevnt to bro fiel of physics n chemistry, e.g. spin chemistry [10]. If one, rme with the Bell inequlities, clims tht there re no Einsteinin hien vribles to explin entnglement, it is necessry tht the mthemtics upon which tht clim is bse is correct representtion of the entnglement physics. We showe, with simple exmple with Gussin norml ensity, tht the mthemtics behin the inequlities is nomlous. References [1] J.S. Bell, On the Einstein Poolsky Rosen prox, Physics, 1, 195, (1964. [] A. Einstein, B., Poolsky, n N. Rosen, Cn Quntum-Mechnicl Description of Physicl Relity Be Consiere Complete?, Phys. Rev. 47, 777, (1935. [3] D. Bohm, Quntum Theory, pp , Prentice-Hll, Englewoo Cliffs, [4] J. von Neumn Mthemtische Grunlgen er Qunten Mechnik, Springer, 193, oi / [5] J.S. Bell, On the problem of Hien Vribles in Quntum Mechnics, Rev. Mo. Phys., 38, , (1966. [6] D. Dieks, Von Neumnn s impossibility proof: mthemtics in the service of rethorics, philscirchive.pitt.eu/vnproof (016. [7] H. Geures, K. Ngt, T. Nkmur & A. Frouk, A note on the possibility of incomplete theory, rxiv , (017. [8] H. Friemn, Unproveble Theorems, Boston Mthemtics Colloqium, MIT October 8, (009. [9] B. Noren, Entngle photons from single toms n molecules, oi/ /j.chemphys [10] Y. Tnimoto n Y. Fujiwr, Effects of high mgnetic fiels on photochemicl rections, in: Hnbook of photochemistry n photobiology, E. H.S. Nlw, vol 1 Inorgnic Photochemistry, Chp 10,
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