imaginary numbers as quantum superposition states and time...

International Journal of Arts and Sciences 3(9): 274-295 (2010)

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Imaginary Numbers as Quantum Superposition States and Time-like Dimension Witold Wiszniewski, Silverbrook Research, Australia Abstract: The paper deals with the imaginary numbers from the perspective of mathematics, physics and philosophy. The core proposition is that the unitary imaginary number i is a double value number with the assigned not one but two different numbers at the same time, These numbers are proposed be: ½ and –½, fact of which is written in the so-called i-equation: i = [½, –½] or i = [–½,½]. Philosophical considerations aim at explanation or justification of such a paradoxical double value assignment, looking for a similar paradoxical at the same time appearances possibly taking place in philosophy and physics. The core of the philosophical analysis is an account of McTaggart’s proof of the unreality of time in which he claims that sequentially occurring events are not only sequential but they also appear at the same time. Then in physics – Special Theory of Relativity Theory (STR) – paradoxical at the same time appearances take place when in its inertial frame of reference the propagated the light ray is in countless number of places at the same time. Similarly in Quantum Mechanics (QM) when a superposition is present the spin of electron can be up and down at the same time as well. We try to make use of these paradoxical at the same time appearances to propose a new model of the time-like dimension of the spacetime, as well as to form a mathematical theory of the imaginary numbers based on an abstract superposition. However although paradoxical, at the same time appearances can in fact constitute a contradiction. But it is argued that introduction of time going backward can disarm this danger, as there is nothing contradictory in the concept of such a time. The paper also touches the history when mathematical inputs of Hamilton and Buée are reflected upon. Keywords: superposition, imaginary numbers, spacetime, time-like dimension 1. Introduction Imaginary numbers appear in an irreducible way in the Special Theory of Relativity and Quantum Mechanics. The thought then is that it might be possible to extract some concepts from these theories which can help us to understand better these numbers. In this respect it could be the STR concept of the time-like dimension of the spacetime and the QM concept of superposition. Both concepts invoke imaginary numbers explicitly or implicitly. Giving preference to QM and considering that quantifications of superposition states necessary involve imaginary numbers we propose to take imaginary numbers as abstract superposition states. While concentrating on imaginary numbers we do not depart from the standard approach with treating these numbers as the part of complex numbers, taken as ordered pairs of real numbers.



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The paper starts with an outline of Hamilton’s theory of complex numbers as it provides some justification of the negative square. The main part of this paper contains the construction of an abstract superposition state of two numbers which is then identified with the unitary imaginary number i. Although the choice of these numbers is partially arbitrary for the reasons outlined before we say that these numbers should be ½ and –½ and we write this in a form of the i-equation: [ ]½ ½, −=i , or [ ],½ ½, −=i where the square brackets mean the possession at the same time of the listed and ordered values, therefore it is a double value number. The choice of ½ and –½ numbers forming the unitary imaginary number i is confirmed by the independent derivations of the exactly the same i-equation from the formalism of QM and also independently from the STR. The latter one allows us also to come up with a new understanding of the time-like dimension and therefore spacetime as well. These two derivations provide the physical underpinning of imaginary numbers. If indeed the unitary imaginary number can be taken as the superposition states of ½ and –½ then metaphorically speaking, or not, such a number is a scalar entity with two values: ½ and –½, assigned to it at the same time, in short a double value number. This is a rather unusual claim which needs further justification. It is proposed then that the common dominator of these physical underpinnings of imaginary numbers is a set of unusual and paradoxical appearance at the same time. In the STR, due to time dilation, in its own frame of reference, the ray of light is in countless different locations at the same. In turn as said before, in QM a spin ½-particle can have the spin up and down at the same time. This clearly leads to general philosophical reflections about paradoxical at the same time appearances. As it happens the central piece of philosophy of time, i.e. McTaggart’s proof of the unreality of time, deals explicitly with contradictory appearances which are sequential in time but also are at the same time as well. The challenge in our endeavour is therefore to avoid such at the same time contradiction. Finally there is an interesting historical background of these considerations, mainly related to the question of what Hamilton’s secondary moments of time are, the moments which play an essential role in his theory of complex numbers. In Hamilton’s words these moments are taken without regarding whether they follow, or coincide with, or precede the primary moments, in the common progression of time. In this historical aspect we also mention that in 1805, French mathematician Buée proposed something similar to the i-equation. Both of these historical aspects receive attention in the paper. 2. Hamilton’s Algebra as the Science of Pure Time Hamilton presents his theory of numbers in his Theory of Conjugate Functions, or Algebraic Couples; with a Preliminary and Elementary Essay on Algebra as the Science of Pure Time.1 This work follows the intuitionism school of the foundation of mathematics. Hamilton rests the theory of numbers on the concept of moments of time. He derives the natural, fractional, rational and real numbers from the so-called primary



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moments of time. The starting point is the formation of ordinal numbers by generation in a thought the primary moments towards the future and towards the past. This forms the ordinal numbers: first (moment), second … Hamilton then defines a step ‘a’ (not italic font) in the progression of time as the difference between two moments: a.=− AB By multiple compounding of the moment A with the positive step ‘a’ and contra-positive step ‘–a’ the series of integers ,3,2,1,0,1,2,3 −−− is created. The formation of all types of numbers up to real numbers follows. For complex numbers, Hamilton introduces the concept of the secondary moments of time described as follows:2 When we have imagined any one moment of time A1, which we may call a primary moment, we might again imagine a moment of time A2 and may call this a secondary moment, without regarding whether it follows, or coincides with, or precedes the primary moment, in the common progression of time; we may also speak of this primary and this secondary moments as forming a couple of moments, or a moment couple, which may be denoted thus, (A1, A2). The question is, what are these secondary moments of time? As the definition says they are not related to primary moments which are our normal moments of time. Later on we will provide some possible explanation of these moments. Now we say after Hamilton that the prime entity of the complex number theory is the moment couple ( )21, AA , defined as the ordered pair of the primary moment A1, and the secondary moment A2. A step-couple ( )21 a,a is taken as a pair of differences between two types of moments of time:

( ) ( ) ( ) ( )a a1 2 1 2 1 2 1 1 2 2, , , ,= − = − −A A B B A B A B (2.1)

which is composed of primary and secondary steps in the progression of time:

( ) ( ) ( )2121 a,00,aa,a += (2.2) Step-couples can be added, subtracted and multiplied by a number:

( ) ( ) ( ) ( ) ( )212122112121 a,aa,aba,bab,ba,a aaa =×±±=± (2.3) A step-couple can be divided by another, provided that for the latter both the primary and secondary steps are not zero. The result of such a division is a number couple, which is the sum of the pure primary number-couple and the pure secondary number-couple:

( )( ) ( ) ( ) ( )2121

21

21 ,00,,a,ab,b aaaa +== (2.4)

It follows that:

( ) ( ) ( ) ( ) ( ) ( ) ( )22122111212121 a,0,0a,0a,aa,a,b,b ×++=×= aaaaaa (2.5) To multiply a pure secondary step-couple and secondary number-couple, Hamilton proposes that:



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( )2222212122 a,a)c,c(=)a,0(),0( aaa γγ=× (2.6)

where γ1 and γ2 are multiplication constants. Hamilton chooses the constants to be:

0and1 21 =−= γγ (2.7) which yields:

( ) ( ) ( )122122112121 aa,aaa,a, aaaaaa +−=× (2.8) Hankins3 points out that this choice is not completely arbitrary since Hamilton continues:

( ) ( )( )212121 a,a,b,b aa= (2.9) It follows:

b a ab a a a

1

2

= += + +

a aa a a

1 1 1 2 2

1 2 2 1 2 2 2

γγ

(2.10)

If for some effective step c: cbcbc,a,ca 22112211 ββαα ==== then:

β α γ αβ α α γ α

1

2

= += + +

a aa a a

1 1 1 2 2

1 2 2 1 2 2 2

(2.11)

The solution of Equation 2.9 in respect of the numbers a1 and a2 is given by:

( ){ } ( )( ){ } 2112

22122112

212221122122111

αβαβαγαγα

αγβγαβαγαγα

−=−+

−+=−+

aaaaa

(2.12)

In order that the numbers a1 and a2 should always be determined by Equation 2.10 when a1 and a2 are not both null steps, it is necessary and sufficient that the factor:

( ) 22

221

2

2212212211 4

121 αγγγααγαγα

+−

+=−+ aa (2.13)

is never null, when α1 and α2 are not null numbers. This condition is satisfied if:

041 2

21 <+ γγ (2.14)

so 01 <γ , meaning that the square of the pure secondary number couple is negative, With these constants the division rule of two step-couples becomes:

( )( )

( )( )

+−

++

== 22

21

211222

21

2211

2211

2211

21

21 ,c,cc,c

a,ab,b

αααβαβ

αααβαβ

ααββ (2.15)

from which the equations for multiplication and division of two complex numbers follow. As the secondary number couple is identified with an imaginary number it follows from

01 <γ that its square is negative assuming that the product of any secondary moment



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couples is given by a primary couple, i.e. .02 =γ This completes our presentation of Hamilton’s theory.

3. Philosophical considerations The main aim of this part is to reflect on what imaginary numbers could be, that is, what philosophical commitments are required to go beyond their formal mathematical definition. To start with, we repeat that imaginary numbers are necessary in the formulation of the STR and QM. In the STR, these numbers appear when the relativistic spacetime is postulated and the postulation is given in the Poincaré representation. In turn, QM uses imaginary numbers to quantify amplitudes of probability which are necessary to calculate outcomes of quantum processes. Likewise, with the negative square of imaginary numbers, the crucial axioms of both theories are beyond our conceptual grasp. The STR presents the Light Principle, according to which the speed of light does not depend on the speed of its source. Although unequivocally true, this principle is unexplainable. Similarly, QM proclaims stochastic outcomes of micro-world processes, the source of which is the superposition of incompatible properties, which is also awkward to understand. Feynman, commenting on QM, famously said4: ‘Do not keep saying to yourself, if you can possibly avoid it, check it “But how can it be like that?” because you will get ‘down the drain’, into a blind alley from which nobody has yet escaped. Nobody knows how it can be like that.’ Probably Feynman is right about the fruitlessness of attempts to understand the axioms of QM. The same might be true with the Light Principle and our negative square as well. But if these impossibilities to understand are connected then there might be some sort of a common denominator of impossibility to understand. What could be this dominator? We claim that it is the occurrence of certain things at the same time, whereas for our mind they must be sequential in time. QM deals with superposition states where, for example, the intrinsic spin of an electron is neither ½ ħ nor –½ ħ but rather the electron has these two values at the same time. In turn in the STR there is Poincaré’s light clock where the light ray’s emission occurs at the same time (the time in the light ray inertial frame of reference) as its detection, although normally these moments are sequential in time. Interestingly, in the philosophy of time there is a century of the debate about these unnatural appearances at the same time. We refer here to McTaggart’s proof of the unreality of time.5 McTaggart, a or in fact the creator of the modern A-type theory of time provides an argument that this theory leads to a contradiction, which involves unnatural occurrences at the same time. The proof engages the fundamental characteristics of the A-type and B-type theories of time. The A-type theory advocates a dynamic time nature, admits the reality of now and also usually time flow as well. According to the B-type theory, time is static, as the relativistic Block Universe Model prescribes, and the moment of now (present) is purely subjective. Both theories use the series of events, times and object stages to formulate their claims. In the A-type theory these are A-type series with ordering according to how much pastness or futurity the elements of the series have, or whether they occur now. In contrast the B-type series are



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ordered according to the later than criteria without any privileged present moment. The B-type theory needs the B-type series only, but A-type theory requires both A-type and B-type series. In McTaggart’s proof usually the series of events is used. The proof goes as follows: 1. Events are located in a B-series (ordered with respect to later than), only if time exists. 2. Time exists, only if there is genuine change. 3. There is genuine change in the world, only if events are located in a real A-series. Therefore: i. Events are ordered with respect to later than, only if they are located in a real A-series. 4. If events are located in a real A-series, then each event acquires the absolute properties past, now, future. 5. There is a contradiction in supposing that any event has any two of these absolute properties. Therefore: ii. A real A-series cannot exist. Therefore: iii. Events are not ordered with respect to later than. iv. A real B-series cannot exist. v. Time is unreal. The proof makes sense only in the frame of the A-type theory of time, because the B-type theory negates the second and the third claims (assumptions). The first three claims establish the A-type theory of time with the crucial progression of an event from the future, through the present, to the past. Then there is Claim 4 which proclaims that each event is in the past, is present and is in the future, we may say ‘at the same time’, that is, in a simultaneous way, not timeless because the proof applies only to the A-type theory of time. But this creates a contradiction as Conclusion (v) states. There are many justifications for Claim 4. Let us consider McTaggart’s original justification when he claims, firstly, that the attribution of the characteristics past, present and future to the terms of any series leads to contradiction, unless it is specified that they have them successively. Simply and crucially given moments of time, events and object stages are exclusively in the future, or at present (now), or in the past. This means that if we find some unnatural occurrences at the same time, being say in the past and in the future, we face the contradiction. Secondly, as Geach explains, if it is true that, say, Queen Anne has died, it is also true from some other temporal perspectives that she is dying and it is also true from another perspective that she will die.6 Similarly Horwich says: ‘The Universe contains the following facts about an event E: E is now, E is future and E is past.ten’7 In both cases these creates the contradiction because there seems to be no temporal perspective from which the two conjunctions of the three (or two) temporal statements can be true. As McTaggart upholds the A-type theory he is forced by this contradiction to declare that time is unreal.



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Although this conclusion does not have general support among A-type theorists of time, McTaggart seems to have point here. As Gale says: ‘Thus McTaggart wins no victory but a stalemate, which is a victory of sorts, given how hopeless his doctrine of the unreality of time initially appeared.’8 If so then still there might be a temporal perspective from which all three temporal statements about Queen Ann’s death are true. This would mean that unnatural occurrences at the same time might have some reality. If time is declared unreal as McTaggart demands, so be it. However we need to ask what kind of temporal perspective can generate occurrences at the same time? It cannot be our subjective perspective since we are seemingly always now without much access to the past or the future. We might try to get some objective perspective from no-when. But as Dummett says we can have a view from no-where when we imagine space, but the view from no-when is impossible.9 We do not see, for example, the time-line of the day’s temperature. Still remaining in the frame of the A-type theory of time, at this point let us treat temporal issues differently. Why do not we construct a view from a time perspective? Imagine that there is a time-wagon with an observer who moves in the same way as time flows. It is clear that time flow is the abstraction from all flows and movements therefore there is neither a reference point nor a clock for such flow. No clock for time flow is a clock which stopped for this flow; therefore it is the flow of time at the same time. So for this observer everything happens at the same time. The above temporal singularity is not as unreal as one might think. Because time is the propagation of light, according to Einstein’s definition, there is a legitimate perspective of time from a time perspective, which is the perspective of the inertial frame of reference moving at the speed of light or asymptomatically closely to it. In this frame all temporally diverse events are co-temporal: they occur at the same time, as long as from a normal perspective they are not infinitely separated. So we have a curious perspective allowing the accounts of Queen Anne’s death and Horwich’s assertions to be understood. The addition of this perspective allows the retention of temporal separations of these assertions, and an assertion of their occurrence at the same time as well. However we can argue as much as we wish, the fact remains that very often these paradoxical appearances at the same time are taken to be plainly contradictory. Steering away from endorsing contradiction, and therefore not subscribing to Dialethism we say that if we introduce time flowing backward then the potential at the same time contradiction will be disarmed. Simply moving in time forward by a given duration and then going back by the same duration brings us to the same moment of time. So for example displacement of the light ray in the Poincare’s clock by ½ ct and –½ ct at the same time can be equivalent to the to the displacement by ½ ct forward in time and ½ ct (that is –½ ct) backward in time. Although unusual formally there is nothing contradictory with the backward time.



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Fig. 1 Poincaré clock Now if indeed there are unnatural occurrences at the same time, or time flowing backward as well then there should be something in mathematics which would quantify them somehow. A reasonable candidate for such a role is a set of imaginary numbers. Independently of anything we can then envisage them as scalar entities which have not one but two different values assigned to them. Again we can call them double value numbers. A natural choice for these values could be 1 and –1. However taking the benefit of hindsight we choose instead ½ and –½ and write this as the i-equation: [ ]½½, −=i , or

[ ]½ ½, −=i . It is important to keep in mind that the listed and ordered values inside the square brackets are possessed at the same time. To emphasize importance of this we call the square bracket with its content the quasi-temporal change. Such a change is defined by Wiszniewski10 as the possession of incompatible properties at the same time, which is a derivative of the definition, attributed usually to Aristotle, where change is taken as a possession of incompatible properties. However the drawback of this philosophical derivation is that liking or not the contradiction is always around the corner. After all the core argument is provided by McTaggart who claims that time is contradictory and we use such a time to construct the theory of imaginary numbers. However again we can remove it by taking the refuge with time forward and time backward. 4. Physical underpinning of imaginary numbers 4.1 Quantum Mechanics In the derivations of the i-equation from QM we will employ the at the same time considerations arising from the superposition of the spin up and down states of electron, or maybe more generally a spin ½- particle. Before we start this presentation we need to explain the concept of quantum superposition. In Quantum Mechanics a given physical property, momentum spin etc. is identified with a so-called state of the system, which is a unit-length vector ,∑= iivcv where iv are components of the vector and ic are constants. This vector is positioned in the so-called Hilbert representation space, the dimensionality



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of which is equal to the number of possible values which the given property can take. In the case of ½-spin particle the dimensionality is two; in the case of momentum of moving quantum object it is infinite. Each component of the vector vi represents a pure state of the system, which in turn represents a possible outcome of the measurement. Such possible outcomes are called eigenvalues of the system. In our case of the spin along z direction we have .2211 vcvcv +=

If 1vv = then the eigenvalue is, say +½ħ (spin up) but if

2vv = then the eigenvalue value is –½ħ (spin down). The scalar inner product of the state vector and a given component of this vector determine amplitude of probability ψ that the measurement returns the eigenvalue associated with this vector component. This amplitude is usually given by a complex number. In turn the probability of the given outcome of measurement (not amplitude of probability) is given by the product ψ ψ* where ψ* is the conjugate of ψ; the product which takes the real value, and as it represents a probability it must satisfy the condition: ψ ψ* .1≤ Now Dirac’s version of definition of the superposition, reprinted for example from Hughes11 goes as follows. If v1 and v2 represents possible pure states of the system, then any vector 22113 vcvcv += such that 13 =v also represent a state of the system. Assuming that 22113 vcvcv += still represents the spin along z direction where 1ν represents say the spin up and 2ν represents spin down we might ask what kind of eigenvalue 3v represents. The answer is that it represents the superposition state of the spin up and the spin down. Metaphorically speaking, or not, it amounts to the spin being up and down at the same time. Incidentally the spin superposition along z direction amounts to spin being up or down along x direction and also it could amount to spin being up or down along y direction. We can see here how different Quantum Mechanics is from classical physics. Classically the spin can be either up or down but not up and down and the same time. However the fact remains that in the superposition the spin is up and down and really nothing changes if we add this occurs at the same time. If this is not sufficient to remove a potential contradiction in the rephrased statement we can say that when spin is up the particle spins, or whatever it does, in time forward and when the spin is down it spins backward in time. However this statement does not much good since we do not really know what the spin ½- particle does when it has spin up or down. Now we say that generally without movement the quantum amplitude of probability does not vary in space but varies in time:

( )/exp~)( iEtt −ψ (4.1) As said before the set of eigenvalues is the set of allowable values of the given physical property. In the case of ½-spin particle for z direction the two eigenvectors are: (1,0) and (0,1) with the eigenvalues ½ħ and –½ħ. These states can be denoted as 1 and

2 respectively. For x direction the eigenvectors, denoted by I and II are: (1, –1)/√2



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and (1, 1)/√2 also with the eigenvalues of ½ħ and –½ħ. Now let us assume that the electron is in the state 1 spin up in z direction. It has the spin ½ħ which arises from the

eigenvalue equation: 01

21

01

1001

21

=×−

where 10

0121

− is one of the three Pauli

spin matrices. The full time dependent state vector is given by:

( ) ( ) /exp1/exp1 tBitEiz µ==Ψ + (4.2) Similarly in the x direction the spin vectors are:

( ) ( ) /expII/expI tBitBi xx µµ =Ψ−=Ψ +− (4.3) where vectors IIandI are given by the equations:

( ) ( )11

2121

21II

11

2121

21I =+=

−=−= (4.4)

Now we create the superposition of IIandI :

( ) ( ) /expII/expIsup tBitBi µµ +−=Ψ (4.5) But at time t = 0 the superposition function needs to be normalized meaning the module of probability amplitude of having two values of spin at the same time must be equal to 1:

( ) ( ) /expII2

1/expI2

1sup tBitBi µµ +−=Ψ (4.6)

Now supΨ is exposed to the σx operator 0110

21 making the weighted state vector Wsup:

∑∑ ==Ψ=22

supxsupi

iii

ii vavPW σ (4.7)

where Pi, ai, and vi are projectors, eigenvalues and eigenvectors of the σx operator. So:

( ) ( ) /expI22

1/expII22

1supxsup tBitBiW µµσ −−=Ψ= (4.8)

Next we create the scalar product SQM of the vector 1 and supW that is sup1 W :

( ) ( )

( ) ( ) ( )

/sin½ /exp¼/exp¼

/expII22

1/expIII22

11 sup

tBitBitBi

tBitBiWSQM

µµµ

µµ

=−−=

−−== (4.9)



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where == II1I1 1/√2. This product is the probability amplitude of the electron being

first in the state 1 and then being in superposition of the states I and II after time dt:

τµ /½/½QM dtidtBiS =≈ or τ/½QM dtiS −≈ (4.10) where τ = ħ/μB is a time constant. Now we have to create SSC, a semi-classical counterpart of SQM, which expresses the fact that the in x direction the spin is up and down i.e. ½ħ and –½ħ at the same time and we write this in the form of the i-equation:

[ ] ½,½ −∝SCS or [ ] ½,½−∝SCS (4.11) To be on par with SQM we should multiply it by the time factor dt/τ:

[ ] τ/½,½ tdSSC −= or [ ] τ/½,½ tdSSC −−= (4.12) Now considering that:

SCQM SS = (4.13) we obtain:

[ ]11, −=i or [ ]11,−=i (4.14) which is close to the i-equation [ ]½½, −=i or [ ]½ ½, −=i but not the same. However have you ever wondered why the spin value of electron is ½ ħ instead of ħ? It seems that the electron rotates in a metaphorical sense two times slower than it should. Factually if the spin analyzer rotates with angular velocity ω then the amplitude of the probability rotates with half of this velocity ½ω. Generally, the former rotation is given by the normal rotation matrix and the latter by the half angel rotation matrix. For example, for a rotation around the x axis we have the real rotation matrix and half angle rotation matrix:

θθθθ

cossinsincos

−=xR

2cos

2sin

2sin

2cos

θθ

θθ

i

iRx = (4.15)

If the amplitude of probability rotates two times slower than the spin analyzer then time in the frame of the spinning electron might run two times slower, which explains why the spin of the electron is ½ħ instead of ħ. If so then in Equation 4.12, instead of the time factor dt/τ, we should use a factor that is two times shorter: dt/2τ. Considering equation 4.13 we obtain:

[ ] ½,½

22−=

ττdtdti or [ ]

½,½22

−=ττ

dtdti (4.16)

which leads second time to the desired i-equations: [ ]½½, −=i , or [ ].½ ½, −=i



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4.2 Special Theory of Relativity To derive the i-equation from the formalism of STR we take the advantage of the interval metric which provides the measure of the relativist spacetime. There are two fundamental representations of the spacetime. The original one given by Poincaré’s represents the spacetime as x, y, z, ict. However because the problem with the conjugation of the complex spacetime vectors Minkowski replaced normal time t by –it and obtained the representation: x, y, z, ct, which today bears his name. It is important to realize that both representations use the imaginary numbers involve imaginary numbers, Poincaré’s explicitly and Minkowski implicitly. In our reasoning we use Poincaré’s representation: x, y, z, ict of the relativistic spacetime. In each inertial frame of reference, the spacetime can have different time coordinates: ict, ict’, ict’’ and we also adhere to A-type theory of time, not very popular today though, with its time flow and reality of now. The question is how we can imagine the time-like dimension that encompasses all moments of time. Such a dimension must be a superposition of all moments and displacements positioned on a space-like line at the same time. At the same time because we take this dimension as it exists now and not what temporally becomes. Consequently the all moments of time cease to be sequential in time and they become contradictorily at the same time, because A-type theory demands that. Light can give us this temporal superposition. If on a straight light path, being a part of some non-light inertial frame of reference, a set of light ray’s spatial positions is marked, then on one hand we have the collection of the ray’s lived through moments at the same time (according to the STR this is the case in the light frame of reference), and on the other this dimensional positioning reflects the sequential temporal account given in the specific non-light inertial frame of reference. Applying this reasoning to the Poincaré’s clock, the clock where light is emitted towards the mirror and comes back in time t, we see that the moments of light’s emission, reflection and detection are not only sequential in time but also occur at the same time. So apart from their sequential occurrence, the light displacements ½ct and –½ct occur at the same time too. Interestingly such co-temporal displacements are equivalent to the displacement ½ct forward in time and the displacement –½ct backwards in time. Consequently the time-like dimension can be given by two oppositely orientated axes. The above alternative replacement at the same time characterization with forward and backward displacements in time is very attractive since in this context at the same time characterisation either suggest or imply contradiction; whereas even in the A-type theory there is nothing wrong logically with time flowing in the opposite direction. Maybe such a time would explicitly exemplify the temporal symmetry of the laws of Physics. Now we can argue that time forward and time backward in a sense amount to be at two different places at the same time because displacement ½ ct forward in time and displacement – ½ct backward in time are opposite temporal displacements covering a non zero ct durations ending up at the initial moment of time ct. In other words each moment of time can enclose a temporal loop of any duration. Obviously in the day to day experience we



286

do not encounter time flowing forward together with time flowing backward that is the symmetric time. However the symmetric time can manifest itself in the paradoxical situations presented by Physics, for example in STR where such a time could be a part of the relativistic spacetime in the same way as Newtonian time is a part of the E3 x E1 classical spacetime. The symmetrical time should also have its place in QM although it would have to be a different formalism would reflect the presence if the complex vector conjugation which is no present in STR. Going back to our argument about the nature of time like-dimension we can provide another justification of the time-like dimension being a composition of two oppositely orientated axes, which probably is valid also in the frame B-type theory of time. If some entity moving with the speed of light emits a light ray along its journey then according to the STR the light ray travels only with the speed of light and its displacement is ct and not 2ct. This can be rephrased by saying that the light is displaced normally by ct, but additionally on top of it, it is displaced unnaturally back and forward by ½ct and –½ct at the same time. Again we can suggest here the alternative time forwards and time backward displacements. These back and forward displacements remove the excess displacement ct as they cancel each other out. This is the so-called Double Arrow Contradiction proposed by Wiszniewski12. A similar argument can be formed when compounding velocities are perpendicular to each other. It is claimed that these cancelations, in other words the displacement back and forward at the same time, or displacement forward in time and backward in time, constitute the time-like dimension. We might have a trouble to convince a broader audience of this metaphysical reasoning. However because it leads to an important finding we ask the reader to accept it just for argument’s sake. So if indeed ½ct and –½ ct occurs at the same time, or ½ct is the displacement forward in time and –½ct is the displacement backwards in time, then [½ct, –½ct] constitute the time-like dimension. Such a dimension must satisfy the interval metric which is an import this metric from STR:

[ ] ( )22½,½ ctctct =− (4.17) therefore:

[ ] ictctct =−½,½ (4.18) To proceed we divide both sides of this equation by ct which leads to the i-equation

[ ]½½, −=i , or [ ],½ ½, −=i for the third time. It is important to note that in this derivation we use explicitly Einstein’s assumption that the reflection of the light ray in the Poincaré’s clock occurs exactly at half of the total time of the light ray’s propagation. This is where the numbers ½ and –½ come from. This provides a clear argument that the numbers ½ and –½ are best suited to form the formal definition of the unitary imaginary number i. This is why we asked for agreement about the correctness of the Double Arrow Contradiction argument. Now we are in a position to give a possible explanation of Hamilton’s secondary moments of time. As said before the time-like dimension is given by a pair of ½ct and –



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½ct displacements, co-temporal or forward and backward in time. We can say that these co-temporal displacements constitute the symmetrical time, which can be called dimensional time, as it is identical to the time-like dimension. This time is given of course by [½ct, –½ct] where the square brackets imply at the same time (or time forward and time backward) displacements. Because of the at the same time nature of this time we can also call it synchronic time. Furthermore we can call our normal time diachronic time, which is given simply by magnitude ct. So now the proposition follows to identify the Hamilton’s primary moments of time with moments of normal time and to identify Hamilton’s secondary moments of time with the moments of symmetric, or dimensional time: that is, the time-like dimension, the difference of which (moments) are quantified by the ict magnitude. Do we have any chance to register these secondary moments of time? Obviously directly we cannot. However one might argue that time dilation which can be experienced by comparing the reading of the clock displaced along some closed loop and with the reading of the stationary clock, is the sign of the presence of the secondary moments of time. Interestingly Bigelow13 in his consideration of imaginary and complex and numbers came up with the conclusion that integer imaginary numbers can be instantiated by the co-temporal, synchronic family relations like brotherhood or cousinhood and the integer numbers by the parenthood (grandparenthood) and childhood (grandson-hood or granddaughter-hood) relations. Not surprisingly Bigelow calls these latter relations diachronic relations. The composition of the diachronic and synchronic relations is then said to be quantified by the integer value complex numbers. Finally as Nahin14 reports Buée claimed in 1805 that: “if t represents future time and if –t represents past time, then the present time is composed of 1½ −×t and 1½ −×− t ”. So let us accommodate this intriguing statement. First of all let us propose that the word and between the above two questions means at the same time, so we form the quasi-temporal variable: [ ]1½,1½ −×−× tt . Secondly as Buée states this bracket formula is the present time so we write Present Time [ ]1½,1½ −×−×= tt . Next we multiply both sides by 1−c , replace 1−c i and obtain: Present Time [ ]tt ½c,½c−= . Now inconsistently we replace Present Time with just t and obtain [ ]ttict ½c,½c−= . Finally dividing both sides of this equation by ct we obtain [ ]½,½−=i which is one of the forms of our i-equation! 5. Imaginary numbers as quantum superposition states As said in the Section 4.1 at the heart of the superposition there is an addition of the state vectors with varying weight factors which generates a new state vector which refers to a physically real measurable situation, for example the superposition of the spin up +½ħ and spin down –½ħ. As there is a one to one correspondence between quantum states and the eigenvalues (spin up corresponds to +½ħ and spin down corresponds to –½ħ) for the



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case of this work we can talk about the superposition of the numbers: +½ħ and –½ħ. However it is important to realize that the superposition of these numbers (eigenvalues), cannot be represented by a set of the superimposed numbers, because when the numbers are superimposed they cease to be normal numbers. As we will see later on the superimposed numbers become imaginary numbers, so their squares are negative. Now we propose that these superimposed numbers are kind of normal numbers which are bonded together in such a way that they become superimposed. What this bonding amounts to is explained below. We start our development with a proposition to explicate a positive real number a as )(a↑ and a negative number –a as )(a↓ . For zero we take )0(↑ or )0(↓ . In the next step we introduce a concept of bonding a number. A number can be bonded to another number from the left, or from the right. To do the bonding we need to specify which side of a number is reactive, i.e. bondable. The specification of bonding is done by using the symbol• . So we have four so-called numerical radicals where positive and negative numbers can be bonded from the left of from the right:

( ) ( ) ( ) •↓↓••↑↑• )(aaaa (5.1) or for the number “a” implicitly present:

•↓↓••↑↑• (5.2) The assignment of the bondable side is asymmetric:

•↑−↑=• •↓−=↓• (5.3) This means that if we change the position of the bonding side the numerical radical is inversed meaning it is multiplied by –1. It is clear then that numerical radicals can change the sign in two ways: by changing the direction of the arrow and by changing the bondable side:

•↑−=•↓↓−•=•↓•↓−=↓•↑•−=↓•

•↓−=•↑↑−•=•↑•↓−↑=•↓•−↑=• (5.4)

We can also see that multiplication of a radical by a negative number is equivocal. For instance •=↑↑•×−1 or .1 ↓•↑=•×− Furthermore if we take numerical radicals as vectors – it will be formally done soon – then any numerical radical rotates two times slower in the sequence of multiplications by – 1. For a normal vector the inverse vector has all its components inversed therefore two times multiplication by –1 cerates the original vector. For a radical four times are required to return it to the original. For example with ↑• we have:

↑•=↓•×−↓•=•↓×−•=↓•↑×−•↑=↑•×− 1111 (5.5)



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Also we can see that two such multiplications: ( )111 +=−×− returns a different radical. To have such a behaviour building parts of numerical radicals ↓↑, and • are not the components of numerical radicals, as such radicals look more like two dimensional number. On this topic it would be instructive to show the real life example of such a two dimensional number-like structure. In Fig 2 the political orientation structure is shown. This structure is two dimensional where one dimension goes along left and right and the other along the libertarian - authoritarian division. Furthermore each of these dimensions is a numerical axis, varying, say, from – 10 to +10 so each component of the orientation can be measured.

Fig 2 Two dimensional model of political orientation If one is, say, a left authoritarian, like for example Stalin was, then to inverse this orientation two times multiplication by –1 is required, but to invert a normal vector one multiplication will do. It seems that numerical radicals can be multiplied by a positive number. They should also be added together. It is then possible that they form a vector space. But because of the difficulty of multiplying radicals by a negative number leading to converting one radical to another, a simple vector space will not do. Instead a more complicated space is required. Based on some similarities the radicals can be grouped in two ways: ↓••↑ , and ↑••↓ , , together with ↓••↑ , and ., ↓••↓ These two groups form the dual correlated vector spaces ( )21,VV & ( )43,VV where V1, V2, V3 and V4 are the atomic vector spaces:

4321 ˆ,;,;,;, VaVVV ∈↓•↓∈↑••↑↑∈••↓∈↓••↑ (5.6) Such a grouping rules out the conversion of radicals by multiplication by –1 since it would be the conversion to a non-existent radical in the given space. The conversion of



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one radical to another in each atomic vector space is also impossible since it would have to go through double multiplication by –1 and each of them is ruled out. It is possible to show that all these atomic vector spaces satisfy all remaining required conditions to form linear vector spaces. Next we define the operation of scalar multiplication of numerical radicals by each other. We will consider all combinations of two radicals irrespectively of atomic vector spaces they belong to. To multiply two radicals we must involve the assignments of the bonding side, but not bonding as such. For the time being, the bonding symbols must be inside of the multiplication compound. The multiplication rules are then proposed to be:

;22

22

aaaa

=↑•×•↓↑=•×•↑

−↓=•×•↑−↑=•×•↓ (5.7)

The inner position of bonding marks is natural, although still arbitrary; consequently they might be outside as well:

;22

22

aaaa

=•↑×↓•=•↑×↑•

−=•↓×↑•−=•↑×↓• (5.8)

If the bonding marks can be inside or outside then they might be mixed as well. To perform mixed multiplication we analyze two equations: •↑=↑•×−1 and •=↓•↑×−1 (Equations 5.5). The double multiplication of the radical ↑• by –1 converts it to the radical •↓ . Taking then 2a=•↑×↑• and replacing the radical ↑• with •↓ we obtain a multiplication with the mixed positions: 2a↑=•×•↓ which leads to the following equations:

2222

2222

aaaaaaaa−=•↑×•↑−=•↓×•↓=•↑×•↓=•↓×•↑

−↑=•×↑•−=↓•×↓•=↑•×↓•=↓•×↑• (5.9)

The square of any radical is then negative and equal to –a2

.2a=•↑•×↑

. The negative sign comes from our natural assertion about multiplication of radicals given by Equations 5.7. For the opposite choice we would have the positive square: This is all what we have to justify the negative square. But there is the way forward using Hamilton’s Equations 2.7 and 2.14 showing that the square of a pure secondary number couple (imaginary number) must be negative. Fixing the sign of one product fixes the signs all products and therefore justifies all equations for the products of the radicals. Fixing the sign of one product fixes the signs of all products and therefore justifies all equations for the products of two radicals. So scalar products of numerical radicals exist. This together with the demonstration that the numerical radicals can be multiplied by a real number proves that they form V1, V2 V3, V4 linear vector spaces over the field of real numbers.



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Clearly ↓•+•↑ is a vector from V1, but it can be something else. We now introduce a symbol ↑↓ which stands for an abstract superposition of two opposite sign numbers a and –a: that is, these numbers bonded together. To learn about ↑↓ we calculate its square. Using a chemical analogy we treat ↑↓ as a chemical compound formed by the reaction:

.↓=↑↓•+•↑ (5.10) This equation becomes the definition of a bonding operation. As per the rules of multiplication of numerical radicals we can calculate ( )2↑↓ :

( ) ( )( )( )

22

22

2

2

aDaCaBaA

DCBA

−=↓×•↓•=−=•↑×↓•=

−↓=•×•=↑−=•↑•×=↑

+++=↑↓

↓×•↓•+•↑×↓+•↓•×•↑•+↑•×=↑↓•+•↑↓•+•↑=↑↓

(5.11)

in sum: ( ) 22 4a−=↑↓ (5.12)

Then for ½=a we have: ( ) 12

−=↑↓ (5.13) which proves that:

12 −=i (5.14) therefore:

( )½=↑↓i or ( )½=↑↓− i (5.15) The derivation of this equation is done using V1 vector space, constituents of which are linearly independent. The question is whether the other two radicals: •↓ and ↑• a from V2 are redundant. After all they form inverse equations to Equations 5.15:

.↑=↓↑•+•↓ (5.16) As per Equations 5.10 for a = ½ we have:

( ) 12−=↓↑ (5.17)

therefore: ( )½=↓↑i or ( )½=↓↑− i (5.18)

We might choose by convention:

( )½=↑↓i and ( )½=↓↑− i (5.19) meaning that:

( ) ( ) ( ) ( )↑•+•=↓−↓•+•=↑ ½½and½½ ii (5.20)



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Although the imaginary number is given by ↑↓ or ↓↑ , such a number should also be given by ↓•+•↑ or ↑•+•↓ simply because earlier we required that the addition of the radicals be commutative: •↑+↓•=↓•+•↑ . So the unitary imaginary number i can be a quantum-like superposition of two numbers: ½ and – ½. But the superposition of any two numbers a and b satisfying the equation:

( ) 12 =− ba (5.21)

forms an imaginary number too, providing that different values are assigned to radicals. For instance we write: ( ) ( )01 ↓+••=↑i and use Equations 5.10 to prove it. As said before the definition of the imaginary number i is a matter of choice of which two numbers to pick to form the superposition providing that Equation 5.20 obtains. We chose ½ and –½. The reason for this is based on the fact that in physical reality there is no superposition of spin states ħ and 0 ħ, but there is: ½ħ and –½ħ. This constitutes the physical underpinning of the nature of imaginary numbers. There are physically instantiated superpositions of more than two numbers which can also be used for this definition. We will deal with them on another occasion, but the definition should be as simple as possible and this is why we take: ( )½=↑↓i or its inverse. As the quantum superposition seem to amount to be at the same time in two mutually exclusive states we say that the imaginary number i has two values ½ and –½ assigned to it at the same time and express it for the fourth time by the i-equation:

[ ]½½, −=i or [ ]½,½−=i (5.22) Again we put a caveat here that more correctly the above two forms of the i-equation express the fact that imaginary numbers are double value numbers. Interestingly, it follows from Equations 5.20 that for any a:

0=↑•+•↓+↓•+•↑ (5.23)

( ) ( ) ( ) ( ) 22222 4a−=↑•+•↓+↓•+•↑ (5.24) Now we consider V3 vector space with •↑ and ↑• as they form the superposition↑↑ :

=↑↑↑•+•↑ (5.25) This is the superposition of two equal numbers. To learn more about it we calculate its square. Doing similar calculations as per Equations 5.10 we obtain: ( ) 02

=↑↑ which makes sense since in physics it is not a superposition at all, or it is the degenerated one. Now let us go back to the question of how many radicals do we need? Equation 5.15 shows that two radicals from V1 are sufficient to form imaginary numbers. Still this equation is ambiguous since we can form positive or negative imaginary number. But Equations 5.18 shows that using the vectors from V2 the inverse equation for i is formed.



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This provides a chance to resolve the issue by convention given by Equations 5.19. So, in a sense all four radicals are required. The argument can be restated in a topological way. It can be seen from Equation 5.10 that •↑ might symbolize the left hand, where the arrow is the hand and the dot shows its inner side. Similarly the numerical radical ↓• can symbolize the right hand coming from the opposite direction forming the left handshake ↑↓ . On the other hand looking at the right equations from Equation 5.16 we can see that ↓↑ symbolizes the right handshake. This situation is shown in Fig 2. As four hands are required to make the left and the right handshake, so all four radicals might be required to provide a full account of imaginary numbers. Fig 3 Left and right handshakes (thumbs indicate bondable sides of the hands) Finally we state that the proposed theory is not in conflict with the standard theory of complex numbers which takes these numbers as the ordered pairs of real numbers. We just say that the imaginary numbers are also ordered pairs of real numbers:

[ ]( )bbac ½,½, −= (5.26) This ends the account of the imaginary numbers theory based on the numerical radicals. Before closing this section we want to present an extension of the proposed model of the imaginary numbers based on two concepts. The first one is a numberness N, which is more or less the same as a number. It is denoted by )(a↓ which varies from –a to 0 and

)(a↑ which varies 0 to +a. The second concept is the sideness S, marked as • , which varies from –a to 0 for the left character, and from 0 to +a for the right character. N and S create the representation space C shown in Fig 4.



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Fig 4 Graphical explanation of the proposed model of imaginary numbers The numerical radials are combinations of equal amount of positive or negative sideness S and equal amount of positive or negative numberness N. There are thus four such radicals. The first two: •↑ )(a , ↓• ; and therefore ( ) ( )↓•+•=↑ ½½i as well, are on the axis W, which is at 45 deg angle in respect to the axis S. The other two: •↓ )(a , ( )a↑• ; and thus ( ) ( )↑•+•=↓− ½½i are on the axis E which is perpendicular to the axis W. The W and E axes are imaginary because they accommodate all numerical radicals square of which are negative. As N and S are real the space C is complex. Importantly though

½̂½ •+•=i and ½½̂ •+•=− i are inverse to each other, so they at 180 deg angle, but they are perpendicular as well. The same holds between N and S: SN −= . It can be argued that this model of the imaginary numbers allows the explanation of the half an angle rotation which was considered in the Section 4.1 and also the complex vector conjugation. Furthermore, using the symmetrical numerical radicals of the form

•↑• together with the normal asymmetrical ones like for example •↑ it is also possible to give the account of the relativistic interval metric. However this goes beyond the scope of this article.

6. Reflection As we can see, the positive and negative halves, i.e.: ½ and keep appearing in the deliberations about imaginary numbers. This supports the definition of the unitary imaginary number i where such a number is taken as a superposition of ½ and –½, and not for example 1 and 0. Neither should it be taken it as a superposition of more than two numbers. In short probably the best definition of such a number is the i-equation. But the transition from the realm of superposition to the i-equations bears some metaphysical burden. Although superposition leads to paradoxical situations it is a well accepted feature of QM. The i-equation, on the other hand, is a mathematical device which might



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pose a contradiction, as clearly for our mind no number should carry the signs + and – at the same time. Something important occurs during this transition. Are we allowed to play with contradiction especially in a discipline so logical as mathematics? However in the defence where the actual at the same time appearances were considered we disarmed the contradiction by taking the refuge in the time forward and time backward combination. On the other hand when at the same time features are really only the figure of speech, as it happens in the case of i-equation we just resort to mathematically acceptable term of the double value number, the term borrowed from the concept of the double value SU2 complex rotation matrices.

References 1. Hamilton, W.R, 1835, “Theory of Conjugate Functions, or Algebraic Couples; with a Preliminary and Elementary Essay on Algebra as the Science of Pure Time”, reprinted in 1967, “The Mathematical Papers of Sir William Rowan Hamilton”, Vol. 3 Algebra, Cambridge, University Press, 2. Ibid, p.76. 3. Hankins, T. L., 1976, “Algebra as Pure Time: William Rowan Hamilton and the Foundation of Algebra”, in “Motion and Time Space and Matter”, Ohio State University. 4. Feynman, R. P, 1965 “The Character of Physics Law”, Cambridge Mass. MIT Press, p. 129. 5. McTaggart, J. M. E., 1908, “The Unreality of Time”, The Mind, Vol. 18. pp 457-458. 6. Geach, P.T., 1979 “Truth, Love and Immortality”, University of California Press, Berkeley and Los Angeles, p. 94. 7. Horwich, P., 1987, “Asymmetries in Time”, MIT Press, Cambridge Massachusetts, p. 46. 8. Gale, R.M., 2002, “Time, Temporality and Paradox”, in “The Blackwell Guide to Metaphysics”, Blackwell Publishers, p. 69. 9. Dummett, M., 1978, “A defence of McTaggart Proof of the Unreality of Time”, in Truth and Other Enigmas, Duckworth, London. 10. Wiszniewski, W.R., 2008, “Time Change and Imaginary Numbers”, VDM Verlag, Dr. Müller, p. 135. 11. Hughes, R.I.G, 1992, “The Structure of and Interpretation of Quantum Mechanics, Harvard University Press., p. 92. 12. Ibid, pp. 197-201. 13. Bigelow, J., 1988, The Reality of Numbers, Clarendon, Oxford, p. 95. 14. Nahin, 1998, P. J., “An Imaginary Tale The Story of 1− ”, Princeton University Press, Princeton, Princeton and Oxford, p.75.

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