Representation of Reality Via Electronic Filter

D.C. Anacker

Dedication: To Parents Edward and Stella Anacker

Abstract: We study analog computation of the wave vector for arbitrary Hamiltonian matrix. Emphasis is placed on implications regarding the many worlds interpretation of quantum mechanics.

1. Introduction

A photograph of Feynman’s office blackboard taken shortly after his death displays the aphorism “one does not totally understand something until one can build it” and also an admonishment to learn every computational technique. Perhaps the great physicist was just chalking up a few addenda for a new edition of Surely You’re Joking, Professor Feynman. On the other hand it may be Feynman really was, in so many words, endorsing the study not only of theoretical and observational cosmology but also engineering cosmology. We can participate in engineering cosmology by considering how to model in the laboratory the universe’s wave vector. Many predecessors have ably shown how to simulate quantum systems on digital or “quantum” computers and doubtless such research will continue to flourish. Comparatively little is known about simulating a wave vector via any kind of classical analog computer.

This discussion will show that a certain electronic filter computes the wave vector at all times given arbitrary initial vector and Hamiltonian matrix. For convenience we consider only two component state vectors - - it will be apparent that the methodology immediately generalizes to arbitrary dimensional state vectors.

Because the filter provides a complete simulation of ly(t) >, although it does not ever exhibit a Copenhagen school style collapse of the wave function it is still potentially a good one to one representation of reality under a many worlds interpretation (MWI) of ly(t) >. Although Hawking has pronounced a non collapsing wave vector trivially obvious, understanding has not reached a consensus on all issues, notably the preferred basis problem and agreement with Born’s rule. Since relevancy of the filter representation of reality intimately depends on that of the MWI, this paper will also address the several problems noted with the many worlds interpretation.

Finally it will be shown how a filter model of the universe impacts three important cosmological conundrums - - interpretation of time, “implausible profligacy” concerns regarding the MWI, and the question of where Hamiltonian /Lagrangian information resides in a particle sparse environment such as obtains during few body scattering or just after the big bang.

2. Filter Simulation of Arbitrary Quantum Mechanical Systems

The fundamental idea goes back to an abbreviate account published on the internet in 2006.^[1]

Recalling that an arbitrary purely sinusoidal voltage can be expressed 2llV ll cos (wt + [phase of V]), where w is its frequency and V = llV lle^{(i[phase of}^V^]) is some complex number, it follows that any purely sinusoidal voltage is also expressible as V eⁱ^w^t +V *e^-i^w^t. For such a voltage we shall hereafter refer to V as its “phasor” ( - - regardless of whether or not this be the same definition of “phasor” used elsewhere in other contexts.)

In Fig 2, if one assumes Vin(t) = V eⁱ^w^t +V *e^-i^w^t, ordinary AC circuit theory coupled with the fundamental voltage gain formula of op-amp inverting amplifiers yields

Vout(t) = {w[{1/(Cr) – (w²L/r)}² + w²] ^-1/2eⁱ^[atan(r/[^w^{L – 1/(C}^w^{)]) +}^p^/2] r_F/r₁}^// V eⁱ^w^t

+ {w[{1/(Cr) – (w²L/r)}² + w²] ^-1/2eⁱ^[atan(r/[^w^{L – 1/(C}^w^{)]) +}^p^/2] r_F/r₁}^//*V *e ^-ⁱ^w^t ,

where ^// is just a label used to distinguish a bracketed quantity from another very similar bracketed quantity appearing a bit later on in the discussion.

Notice that the circuit in Fig 2 converts a pure sinusoidal input voltage to another pure sinusoidal voltage output. The phasor corresponding to the output voltage in Fig 2 is manifestly equal to {w[{1/(Cr) – (w²L/r)}² + w²] ^-1/2eⁱ^[atan(r/[^w^{L – 1/(C}^w^{)]) +}^p^/2] r_F/r₁}^// multiplying the input voltage’s phasor. By inspection of this {…}^// multiplicative factor, we see that we can give it arbitrary amplitude by varying the choice of resistances r_F and r₁, and also arbitrary phase ranging between zero and p radians, by appropriately choosing r, L, and C.

In Fig 1, if one assumes Vin(t) = V eⁱ^w^t +V *e^-i^w^t, ordinary AC circuit theory coupled with the fundamental voltage gain formula of op-amp inverting amplifiers yields

Vout(t) = {w[{1/(Cr) – (w²L/r)}² + w²] ^-1/2eⁱ^[atan(r/[^w^{L – 1/(C}^w^{)]) + 3}^p^/2] r_F/r₁}^/ V eⁱ^w^t

+ {w[{1/(Cr) – (w²L/r)}² + w²] ^-1/2eⁱ^[atan(r/[^w^{L – 1/(C}^w^{)]) + 3}^p^/2] r_F/r₁}^/*V *e ^-ⁱ^w^t ,

where ^/ is just a label used to distinguish a bracketed quantity from {…}^// encountered earlier.

Notice that the circuit in Fig 1 converts a pure sinusoidal input voltage to another pure sinusoidal voltage output. The phasor corresponding to the output voltage in Fig 1 is manifestly equal to {w[{1/(Cr) – (w²L/r)}² + w²] ^-1/2eⁱ^[atan(r/[^w^{L – 1/(C}^w^{)]) + 3}^p^/2] r_F/r₁}^/ multiplying the input voltage’s phasor. By inspection of this {…}^/ multiplicative factor, we see that we can give it arbitrary amplitude by varying the choice of resistances r_F and r₁, and also arbitrary phase ranging between p and 2p radians, by appropriately choosing r, L, and C.

The foregoing analysis of Fig 1 and Fig 2 may appear overburdened with formulae, but actually only facts have been recited which are necessary and sufficient to prove the following

Main Lemma

For any complex number R, one can build a “phasor multiplier” circuit (denoted by the diagram in Fig 3) such that any sinusoidal input voltage produces a sinusoidal output voltage obeying the relation [output voltage phasor] = R [input voltage phasor].

With the aid of the Main Lemma one may proceed to analyze circuitry shown in Fig 4. The two op amps on the left in Fig 4 are each wired in “voltage follower” configuration. A voltage follower transmits the same voltage from input to output terminal while drawing essentially zero current through its input terminal.^[2] A voltage follower thus allows voltage connection to be made without modifying behavior of (,i.e., “loading”,) preceding stage electronics. All the circuitry to the right of phasor multipliers R₁₁, R₁₂, R₂₁, and R₂₂ comprise two identical “op amp non inverting adders” each of which add together input terminal voltages. We are interested in predicting what voltages one gets out of Fig 4’s circuitry if one applies sinusoidal voltages to its two input terminals.

Assuming Vin₁(t) = V in₁ eⁱ^w^t +V in₁*e^-i^w^t, Vin₂(t) = V in₂ eⁱ^w^t +V in₂*e^-i^w^t, the Main Lemma implies

Vout₁(t) = R₁₁V in₁ eⁱ^w^t + R₁₁*V in₁*e^-iwt + R₁₂V in₂ eⁱ^w^t + R₁₂*V in₂*e^-iwt ,

Vout₂(t) = R₂₁V in₁ eⁱ^w^t + R₂₁*V in₁*e^-iwt + R₂₂V in₂ eⁱ^w^t + R₂₂*V in₂*e^-iwt .

The last several equations immediately yield

Vout₁(t) = [R₁₁V in₁ + R₁₂V in₂]eⁱ^w^t + [R₁₁*V in₁* + R₁₂*V in₂*]e^-iwt ,

Vout₂(t) = [R₂₁V in₁ + R₂₂V in₂]eⁱ^w^t + [R₂₁*V in₁* + R₂₂*V in₂*]e^-iwt .

The last two equations imply Vout₁(t) and Vout₂(t) are each pure sinusoids and furthermore

Vout₁ = R₁₁V in₁ + R₁₂V in₂ ,

Vout₂ = R₂₁V in₁ + R₂₂V in₂ .

The last two equations obviously can be rewritten in matrix form Vout = RV in wherein 2x1 vectors Vout = (Vout₁,Vout₂ ), V in = (V in₁,V in₂ ), and 2x2 matrix

R is given by R₁₁ = phasor multiplier R₁₁, R₁₂ = phasor multiplier R₁₂, etc. It will prove salient to remember that (according to the Main Lemma) each of the four phasor multipliers in Fig 4 can be constructed so that all four matrix elements of R are totally arbitrary complex numbers.

It turns out there is only a little analysis remaining. Hereafter it will be convenient to refer to the circuitry pictured in Fig 4 simply as filter R. As previously mentioned, the input terminals of filter R draw no current, and hence filter R can be cascaded indefinitely without degradation. In other words, if one connects any integer N such filters together by soldering output terminals of one R filter to input terminals of another, and so on, the result will be a filter describable by relation Vout = R^NV in.

Let us assume that a great many electronically identical R filters have been fabricated and so connected, each of length DX from input to output, and such (long) filter cascade is oriented parallel to a one dimensional “X” coordinate axis. We can now imagine applying a pair of sinusoidal input voltages to one individual filter R at location Xo and measuring the output voltages appearing at another individual filter R located at position X. Our prior formula Vout = R^NV in would be applicable to this situation but in an analogous form, V (X) = R^{(X – Xo)/(}^DX)V (Xo).We have previously demonstrated and remarked that it is possible to construct filter R such that matrix R is completely arbitrary. Accordingly, it is no problem to construct filter R such that matrix R = e ^{– (i
/ h)}^D^{X H}, where H is an arbitrary 2x2 Hermitian matrix. If we now assume that filter R is indeed so constructed, it immediately follows that a cascade of such filters oriented along coordinate axis X obeys V (X) = e ^{– (i / h) H (X – Xo)} V (Xo) which is of course the Heisenberg equation in units such that distance /time is dimensionless.

3. Evidence for Born’s Law Without Wavefunction Collapse

By percent composition of the universe observers are insignificant , and some authors, for example Vaidman^[3], enunciate a many world’s interpretation affording no special role for observers. But in line with the infinite number of ways to expand a vector space it is possible to also express ly(t) > via bases which do single out observers, Everett’s work introducing the “relative state” being a seminal example.^[3,4] Suppose an observer O arranges to measure the z component of spin for a spin ½ single particle system S, (alh >+ bl i>)_S . The quantum mechanical evolution of this process according to Everett would be^[4]

Eq. (EI):

UlM₀ >_O (alh >+ bl i>)_S = alM₀,”h“>_Olh>_S + blM₀,”i“>_O li>_S

where M₀ denotes O’s memory prior to measurement and parentheses the fact that system O records rather than possesses up or down spin. If one furthermore interprets each unity normalized state l… >_Ol…>_Son the right hand side as a separate and equal reality, i.e. “world”, one has in a nutshell the MWI according to Everett.

In terms of pointing the way to various versions of the MWI Everett’s relative state formalism has had undeniable impact. On the other hand it is not difficult to anticipate analyses for which this formalism is inconvenient or in error. For example frequently detection systems seriously disrupt or annihilate intercepted particle states, e.g. particle multiplier tubes such as channeltrons or channel plates; in such cases a relative state expansion akin to the above is too simplistic because it is predicated upon the assumption that the observer, and the rest of the world, have negligible influence on the observed system during the period of interest. However, by the time a multiplier tube based detection system has finished registering a particle of interest said particle has triggered a complex cascade of interactions and is characteristically not in its predetection state. Generally speaking, if one waits long enough the external world violates the type of weak coupling which is a precondition for relative state entanglement. To cite another example, relative state analysis of a series of observations on independent systems presumably requires carrying along an ever burgeoning load of entanglements- - i.e., one for each observed system - - in the calculations,

But most contention surrounding the MWI by way of Everett has not involved these problems but rather controversy over whether prototypical equation (EI) leads to agreement with Born’s rule and also something called the preferred basis problem.

A well known objection to the approach typified by equation (EI) is that it doesn’t appear to exhibit empirically correct Born statistics there being, on the right hand side, one eigenvector with plus spin and one with negative spin. If each of these normalized eigenvectors corresponds to a single real world, it would seem that the counting is off from Born’s prescription.

Vaidman postulates a “measure of existence” taking care of the problem.^[3] In the case of the right hand side of equation (EI), for example, the measure of existence would be aa * and bb* for worlds lM₀,”h“>_Olh>_Sand lM₀,”i“>_O li>_S,respectively.( The possibility of interpreting ly(t) > as an ensemble of worlds is made more compelling (at least to this author) if it can be explicitly recast as a sum of worlds/eigenvectors together with the right multiplicities, i.e., measures of existence. ly(t) > in the prototypical equation (EI), for example, can be recast on the right hand side as

Na*a{lM₀,”h“>_Olh>_S /Na*}+ Nb*b{lM₀,”i“>_O li>_S/Nb*}, eigenvectors and measures appearing inside and outside the brackets respectively.)

Neither of the two types of “world” in the foregoing example would seem to contain internal evidence of Born statistics although a hypothetical external observer is able to detect compliance by comparing relative measures of existence. So according to some^[4] the MWI is not out of the woods until one more explicitly illustrates how each world comes to internally possess records exhibiting such statistics.

One obvious way of accomplishing this would be to set about calculating the statistics of eigenvalues pertaining to some large number of worlds generated by repeated measurement of an observable.

There have been density matrix derivations purporting to accomplish this objective ^[5], but it is not clear that the right issue is addressed in which the same prepared state l ...>_S is measured for which the density operator is always trivially the same, l ...>_S _S<…l.

Before doing a specific example calculation it is helpful to introduce notation and identify general assumptions. We imagine some lengthy series of integer M identical measurements, each on identically prepared state g₁ lv₁> + g₂ lv₂> + g₃ lv₃> +… (,wherein lv₁>, lv₂>, lv₃>,… denote the eigenvectors of whatever operator corresponds to said measurement,) with the object of ascertaining whether or not the MWI implies an observer will indeed measure eigenvalue v_{i ,}as Born proscribed, g_ig_i*M times. During this procedure the observer perceives a sequence of M eigenstates …lv_i>, lv_j>, lv_k>, … and records this at the same number of locations by means of record keeping states. For example, magnetic tapes or discs afford a collection of magnetizeable domains which can be “written” so any conceivable collection of independent records is storable at different locations on the tape or disc.

It is conventional to denote observed states and corresponding record keeping states by {l v_i >}, and {l”v_i”>}, respectively.We can assume the latter set is orthonormal. If observation of an eigenstate l v_i > is subsequently recorded through establishment of the record keeping state l”v_i”> at some location in memory, M - fold sequential measurements on identically prepared systems give rise to some memory state …l”v_i”> l”v_j”> l”v_k”> … comprised of a product of M individual record keeping states each centered at a different site. It follows that such memory product states will be normalized and any two will be orthogonal unless associated with completely identical sequences of eigenvalues.

With these preliminaries one may now illustrate record formation in compliance with Born for the case of M successive measurements each upon an identically prepared (, single spin,) system alh >+ bl i>.

By ly(t) > let us denote the wavevector of the universe, assumed to evolve according to

l y(t) > = U(t,t₀) ly(t₀) > wherein U(t,t₀) = e^{[-i/hH(t-to)]}. Assume the observer to be a human or programmable machine which intends to make M successive measurements of identically prepared system a lh> + b li>. It is certainly not contrarian to denote ly(t₀) > = lI^Sz_M,M₀ >_O ly(t₀) >_R where the first ket in said product represents the “observer” O and I^Sz_M and M₀ represent, respectively, eigenvalues specifying the intent of the human or machine observer to perform z-spin measurement on M successive identically prepared systems a lh> + b li>, and the memory stored at time t₀. ly(t₀) >_R denotes the remainder of the universe at time t₀. Lastly, let us assume a characteristic time t for the observer to set up and then measure z-spin for each of the M-fold experiments. With these notational definitions in hand it is entirely within conventional understanding to assert

ly(t₀+t) > = U(t₀+ t,t₀) lI^Sz_M,M₀ >_O ly(t₀) >_R implies ly(t₀+t) > = a lI^Sz_M,M₀,”h“>_O ly >_R + b

lI^Sz_M,M₀,”i“>_Oly>_R .The second of the immediately preceding equations is just a mathematical restatement of the conventional wisdom saying if we start out with

the intention of measuring spin-z on an initial state a lh> + b li>, after the requisite set up time t our observational equipment will either be in state lI^Sz_M,M₀,”h“>_O

or lI^Sz_M,M₀,”i“>_O with respective probabilities lal² and lbl². (From here on, all h, i symbols within O subscripted kets should be understood to be bracketed

by quotation marks which are, however, for convenience omitted.) It doesn’t appear unconventional to include the multipliers ly >_R in this second equation

either. They may actually be different for the two terms in the second equation,

but they are both unity normalized. In the end we will see that the desired result only depends

on assuming both ly >_R to be unity normalized, not whether or not they are otherwise equal.

For this reason we can adopt the notational convenience of everywhere using the same symbol

ly >_R.Combined with the identity U(t₀+ t,t₀) = U(t,0), the immediately previous two equations

yield

Identity $: U(t,0) lI^Sz_M,M₀ >_O ly(t₀) >_R = a lI^Sz_M,M₀,h >_O ly >_R + blI^Sz_M,M₀,i >_O ly >_R

One can use Identity $ iteratively to obtain ly(t₀+2t) >, ly(t₀+3t) >,…, ly(t₀+Mt) >. For example, since ly(t₀+2t) > = U(t,0) ly(t₀+t) >, our previous formula for ly(t₀+t) > yields ly(t₀+2t) > = a U(t,0) lI^Sz_M,M₀,h >_O ly >_R + b U(t,0)lI^Sz_M,M₀,i >_O ly >_R .

(Identity $ with M₀g M₀, h)

a U(t,0) lI^Sz_M,M₀,h >_O ly >_R = aa lI^Sz_M,M₀,h,h >_O ly >_R + ablI^Sz_M,M₀,h,i >_O ly >_R

(Identity $ with M₀g M₀, i)

b U(t,0) lI^Sz_M,M₀,i >_O ly >_R = ba lI^Sz_M,M₀,i,h >_O ly >_R + bblI^Sz_M,M₀,i,i >_O ly >_R

Combining the last three equations yields

ly(t₀+2t) > = aa lI^Sz_M,M₀,h,h >_O ly >_R + ablI^Sz_M,M₀,h,i >_O ly >_R + ba lI^Sz_M,M₀,i,h >_O ly >_R + bblI^Sz_M,M₀,i,i >_O ly >_R

(proceeding in like fashion)

ly(t₀+Mt) > = aa...alI^Sz_M,M₀,h,h,…,h >_O ly >_R +

(M a) (M h)

aa...ablI^Sz_M,M₀,h,h,…,h,i >_O ly >_R +

(M-1 a) (M-1 h)

bb...blI^Sz_M,M₀,i,i,…,i >_O ly >_R

(M b) (M i)

Let us denote the last enumerated state ly(t₀+Mt) > = Sc_i l i >_O ly >_R, wherein i runs from 1 to 2^Mand Sc_ic_i* = 1. Interpreting ly(t₀+Mt) > not according to the Copenhagen but rather the many world interpretation, one says that the initial state ly(t₀) > has evolved to state ly(t₀+Mt) > which corresponds to an ensemble of N coexisting realities wherein a portion numbering Nc_ic_i* have determinate records corresponding to the eigenvalues associated with state l i >_O ly >_R. The requirement that the many worlds picture of Everett et.al. is one in which Born statistics are in evidence to observers within the worlds boils down to showing that as the number of measurement trials M goes to infinity, the fraction of said N ensembles exhibiting aa*M number of “h” records approaches one.

Since ly(t₀+Mt) > exhibits M!/[U!(M-U)!] different l … >_O ly >_R terms within which the O ket portion contains U up arrows and M-U down arrows, and the coefficient for each such term is a^Ub^M-U, we therefore conclude that end state ly(t₀+Mt) > represents an N-fold ensemble in which NM!/[U!(M-U)!] a^Ub^M-U (a^Ub^M-U)^* “worlds” record U “spin up” and M-U “spin down” on spin measurement of M successive identically prepared systems a lh> + b li>. To agree with the usual Born statistics boils down to seeing if [ M!/[U!(M-U)!] a^Ub^M-U (a^Ub^M-U)*]l_U ₌ _M_aa_* equals one as M goes to infinity.A little bit of algebra including aa* + bb* =1 and Sterling’s large # identity, ln(#!) = #ln# - # establishes said limiting behavior by showing the natural logarithm of [ M!/[U!(M-U)!] a^Ub^M-U (a^Ub^M-U)*]l_U ₌ _M_aa_* goes to zero as M goes to infinity.

Several additional comments are timely. The analysis is almost identical and reaches the same conclusion if we allow for an overall phase change in

a lh> + b li> for one or more of the M measurement trials. A more general proof of agreement between many world and collapse statistics would consider repeated measurement of some observable having arbitrary many eigenstates, not just two. However it is the author’s belief that a very similar analysis would go through. Whether or not Identity $ requires further motivation to that presented above is largely the province of individual mindset. We may as well humor relative state adherents by conceding an Everettian rationale for Identity $ also exists. Without repeating the various caveats, provisos, and other fine details, the derivation proceeds as follows:

U(t,0) lI^Sz_M,M₀ >_O ly >_R

= U(t,t₁) {U(t₁,0) lI^Sz_M,M₀ >_O ly >_R}

= U(t,t₁) { lI^Sz_M,M₀ >_O (alh> + bli>)_S ly >_R^/}

= {a lI^Sz_M,M₀, ”h“ >_O lh >_S + b lI^Sz_M,M₀, “i“ >_O l i>_S} ly >_R^/

= a lI^Sz_M,M₀, ”h“ >_O lh >_S ly >_R^/ + b lI^Sz_M,M₀, “i“ >_O l i>_S ly >_R^/

= a lI^Sz_M,M₀, ”h“ >_O ly >_R^// + b lI^Sz_M,M₀, “i“ >_O ly >_R^///

which comprises Identity $ if as before one denotes immaterially different factors ly >_R^//

and ly >_R^/// by ly >_R.

4. Comments On The Preferred Basis Problem

Imagine that simultaneous to O’s z – spin measurement which happen every interval t , some other observer O* performs an analogous series of measurements each lasting a different time interval t*. We have seen that O’s measurements split the universe up into 2^M different types of world within time period (0, Mt).During the same time period O*’s measurements split the universe up differently into 2^M^t/^t* worlds. One is led to conclude that the universal wave function can be simultaneously expanded as an ensemble of worlds in different but equally real ways. Unless there is some a priori evidence that only one assortment of worlds can exist, objection to this state of affairs has questionable foundation.

5. Advantages of a Filter Universe Model

Starting with the inception of quantum mechanics people have continually hypothesized underlying models. Perhaps a plurality of physicists now suscribe to some variant of a many worlds interpretation.

Among the impressive merits of a MWI is that it removes randomness and necessity for a Copenhagen collapse mechanism from physics.

Comparatively little has been said concerning if or what the MWI can illuminate regarding three other conundrums - - the question of whether the MWI necessarily is, as some have said, implausibly profligate in worlds; interpretation of time; and the question of where Hamiltonian/Lagrangian information resides in a particle sparse environment such as obtains during few body scattering or just after the big bang.

Because the proposed electronic filter can fully and continuously simulate ly(t) > for arbitrary Hamiltonian and initial state, ipso facto, one may consider it a model of the universe under the MWI rubric. According to the MWI the universe’s wave function continually bifurcates into growing numbers of worlds as time progresses, thereby giving rise to a so called “implausible profligacy” criticism.

If the wavevector for the universe ly(t) > can be approximated by a very large N² dimensional vector in Hilbert space then its filter model can be pictured as a stack of square circuit boards each supporting an identical configuration of filtering electronics sandwiched between NxN arrays of connective ports. Port voltages at different layers in the stack give the components of ly(t) > at different times. As time progresses ly(t) > may indeed acquire an exceedingly baroque mathematical complexity but the filter upon which it may be expressed in terms of voltages does not. The filter model of the universe thus serves as a counter example to allegations the MWI mandates ever burgeoning complexity.

The proposed filter model exhibits quantum behavior if certain analogies are noted, among these depth within the filter corresponding to time, and port, i.e. electrical terminal, voltages corresponding to wave vector components. In this model the wave function propagates across a medium, the filter, from which it is distinct. Hamiltonian/Lagrangian information resides within the filter medium rather than the wave function, modeling why physical laws can evidently remain in force in a particle sparse environment such as obtains during few body scattering or just after the big bang.

Time denotes location within the filter medium, a property fundamentally external to whatever voltages - - again analogous to wave functions - - are propagating. This is consistent with quantum mechanics dogma that time parameterizes the wave vector, yet having no corresponding Hermitian operator, is not measureable by that part of the wave vector called “us”.

__________________________

^[1]

http://www.geocities.com/danacker365/fundqm2.html

^[2]

In principle, arbitrarily small input current can be achieved through increasing op amp input resistance. One may simultaneously avoid degrading signal/noise by cooling, as Johnson noise is proportional to the square root of the product Kelvin temperature times resistance.

^[3]

Many – Worlds Interpretation of Quantum Mechanics

(http://plato.stanford.edu/entries/qm-manyworlds/)

^[4]

Everett’s Relative – State Formulation of Quantum Mechanics

(http://plato.stanford.edu/entries/qm-everett)

^[5]

Many – worlds interpretation

(http://en.wikipedia.org/wiki/Many-worlds_interpretation)