Hypercomplex Algebras and their application to the mathematical formulation of Quantum Theory
aa r X i v : . [ qu a n t - ph ] J un Hypercomplex Algebras and their application to the mathematicalformulation of Quantum Theory
Torsten Hertig I1 , Philip H¨ohmann II2 , Ralf Otte I3 I tecData AGBahnhofsstrasse 114, CH-9240 Uzwil, Schweiz [email protected] [email protected] II info-key GmbH & Co. KGHeinz-Fangman-Straße 2, DE-42287 Wuppertal, Deutschland [email protected] March 31, 2014
Abstract
Quantum theory (QT) which is one of the basic theories of physics, namely in terms of E
RWIN S CHR ¨ ODINGER ’s 1926wave functions in general requires the field C of the complex numbers to be formulated.However, even the complex-valued description soon turned out to be insufficient. Incorporating E INSTEIN ’s theory ofSpecial Relativity (SR) (S
CHR ¨ ODINGER , O
SKAR K LEIN , W
ALTER G ORDON , 1926, P
AUL D IRAC
IRAC equation using pairwise anti-commuting matrices. However, a unitary ring ofsquare matrices is a hypercomplex algebra by definition, namely an associative one. However, it is the algebraic propertiesof the elements and their relations to one another, rather than their precise form as matrices which is important. Thisencourages us to replace the matrix formulation by a more symbolic one of the single elements as linear combinationsof some basis elements. In the case of the D IRAC equation, these elements are called biquaternions, also known asquaternions over the complex numbers.As an algebra over R , the biquaternions are eight-dimensional; as subalgebras, this algebra contains the division ring H of the quaternions at one hand and the algebra C ⊗ C of the bicomplex numbers at the other, the latter being commutativein contrast to H . As it will later turn out, C ⊗ C contains several pure non-real subalgebras which are isomorphic to C , letting bicomplex-valued wave functions be considered as composed from facultatively independent quasi-complex-valued wave functions.Within this paper, we first consider briefly the basics of the non-relativistic and the relativistic quantum theory. Thenwe introduce general hypercomplex algebras and also show how a relativistic quantum equation like D IRAC ’s one canbe formulated using hypercomplex coefficients. Subsequently, some algebraic preconditions for some operations withinhypercomplex algebras and their subalgebras will be examined. For our purpose, an exponential function should be ableto express oscillations, and equations akin the S
CHR ¨ ODINGER ’s one should be able to be set up and solved. Further, likewithin C , functions of complementary variables (such like position and momentum) should be F OURIER transforms ofeach other. All this should also be possible within a purely non-real subspace. It will turn out that such a subspace alsomust be a sub algebra , i.e. it must be closed under multiplication. Furthermore, it is an ideal and hence denoted by J .It must be isomorphic to C , hence containing an internal identity element . The bicomplex numbers will turn out to fulfilthese preconditions, and therefore, the formalism of QT can be developed within its subalgebras. We also show that thebicomplex numbers encourage the definition of several different kinds of conjugates. One of these treats the elementsof J precisely as the usual complex conjugate treats complex numbers. This defines a quantity what we call a moduluswhich, in contrast to the complex absolute square, remains non-real (but can be called ‘pseudo-real’). However, we donot conduct an explicit physical interpretation here but we leave this to future examinations. keywords algebras, bicomplex, hypercomplex, quantum mechanics, quantum theory, quaternions, S CHR ¨ ODINGER equa-tion, special relativity, wave functions.
The history of quantum theory starts with the discovery of the wave-particle-dualism of light (in the broadest sense) byM AX P LANCK (explanation of black-body-radiation, 1900) and A
LBERT E INSTEIN (explanation of the photoelectriceffect, 1905). It means that electromagnetic radiation of frequency ν respectively the pulsatance (angular frequency) ω = 2 πν can be absorbed or emitted in ‘portions’ or quanta of E = hν = ~ ω only, where h is P LANCK ’s constant (orquantum of action) and ~ = h π ≈ , × − Nms is called reduced P
LANCK ’s constant or
Dirac ’s constant. Thisdualism, however, is not confined to electromagnetic radiation: Searching for a plausible explanation for the stability ofelection states within an atom, L
OUIS V ICTOR DE B ROGLIE applied this dualism to matter in 1924, postulating that toany particle of energy E and momentum ~p , a pulsatance ω = E ~ and the wave vector ~k = ~p ~ can be attributed. The wave equation and its complex ansatz for a solution E RWIN S CHR ¨ ODINGER seized DE B ROGLIES idea in 1926.He replaced the classical variables by differential operators to develop one of the most important basic equations ofquantum mechanics (QM), the wave functions φ ( ~x, t ) being its solutions. The general real solution a cos( ~k · ~x − ωt ) + b sin( ~k · ~x − ωt ) , a, b ∈ R (1)turned out as unable to solve the equation not least because it is of 1st order in time derivative which requires some kindof exponential function to solve it. Due to L EONHARD E ULER ’s formula e iϕ = cos( ϕ ) + i · sin( ϕ ) , the complex-valuedansatz ze i ( ~k · ~x − ωt ) = ze i ~ ( ~p · ~x − Et )) , z ∈ C . (2)turns out to be apt because it unifies the trigonometric functions with exponential functions and thus solves linear differ-ential equations of different orders including the 1st and the 2nd. Interpretation of the wave function
Not least because of their complex (hypercomplex, respectively) values, a livelydebate on the nature of these wave functions soon arose. S
CHR ¨ ODINGER considered them as representations of physicalwaves at that time, he thought e.g. of a distribution of charge density.However, the majority of physicists disagreed. Within the year, M AX B ORN suggested an interpretation for the absolutesquare of the wave function as the probability density which is still valid today. This lead to the Kopenhagen interpretationwhich in some aspects seem akin to positivism. Its most famous proponent N
IELS B OHR regarded the wave function asnothing but a useful mathematical aid without any physical reality.In this point, we disagree. We consider complex-valued functions to have special physical properties and being much morethan just a mathematical aid [16], for we are convinced that no such aid or pure formalism could have real physical effects,e.g. in form of destructive interference requiring the wave functions themselves to interfere, not simply the probabilities.This quantum realism also holds for hypercomplex approaches, these obviously being inevitable for a consistent depictionof nature. Note that it is false to identify real values with measurable and imaginary values with not measurable ; the realpart of a wave function is as little measurable as its imaginary part. Actually, the only thing to measure are eigenvalues ofHermitian operators; however, it is possible to reconstruct probability densities, i.e. the absolute squares discussed aboveby multiple measurements on identically prepared quantum systems. It is the phase which remains unknown. Special relativity and hypercomplex extensions
Roughly at the same time as QT, the special relativity theory (SRT,E
INSTEIN , 1905, see appendix C) came to existence as an offspring of the cognizance that like to the laws of mechanics,G
ALILEI ’s principle of relativity also applies to J
AMES C LERK M AXWELL ’s electrodynamics which implies that c =299792458 ms , the vacuum speed of light and other electromagnetic waves has to be the same in any inertial system, beingindependent of its velocity. Involving SRT in wave mechanics, the scalar complex ansatz turned out to be insufficient forthe purpose of fully describing matter. The problem was solved by P AUL D IRAC in 1928 by setting up an equation with hypercomplex coeffizients. These are written as quadratic matrices, while the equation’s solution are vectors of functions.
Conventions for the following text
Universal constants like c or ~ are actually artifacts of the measuring system (seeappendix C) and don’t reveal anything deeper about mathematical relations. Therefore theoretical physicists prefer naturalunits in which they are equal to unity or at least a simple dimensionless number. So we do, using a system of measurementwith ~ = 1 , c = 1 unless an exception is explicitly indicated. So, (2) becomes ze i ( ~p · ~x − Et ) . (3)In conformity with the conventions of relativity theory, especially general relativity, we further use Greek indices if theset of indices includes zero and Latin ones otherwise. Double indices, especially when one of them is an upper (not to be As relativistic QT shows, a real equation with special real × -matrix coefficients and a real wave function 2 component vector as a solutionwould work as well, though less elegant. However, the coefficients then were isomorphic to complex numbers. An exception may be some special states in photons known as coherent states; in this case, the number of ‘particles’ is not sharply defined. this is the today value which is exact by definition since the redefinition of the meter by GCPM in 1983, being within the last error (1973). confused with powers!) and one is a lower, will be summed over unless explicitly negated. Integrals without bounds arenot to be taken as indefinite but as improper , i.e. the integration is to be calculated over the entire range of the integrand.Last, we write operators of the form ∂∂x , ∂ ∂x , ∂∂t , ∂ ∂t , · · · in a space-saving manner like ∂ x , ∂ x , ∂ t , ∂ t , . . . unless there is anyway a fraction. QT is formulated in two manners which look profoundly dissimilar at the first sight: matrix mechanics (W
ERNER H EISENBERG et al., 1925) and wave mechanics (E
RWIN S CHR ¨ ODINGER et al., 1926). S
CHR ¨ ODINGER , indeed, provedboth manners as equivalent [20, 21, 22].Matrix mechanics is more general and coordinate-independent . It deserves primacy in respect of that any wave mechanicshave to be expressible in terms of matrix mechanics , and it provides all concepts and formalism described in appendixB.1. In B.2, a two-state-system and a space of wave functions (in position representation) are shown as two mostly dif-ferent examples of H ILBERT spaces, i.e. spaces of quantum states. Wave mechanics is hence a special case of matrixmechanics. However, it is more graphic since it describes a “particle” by functions in space and time. Additionally, itpromotes the usage of complex-valued functions which correspons to our purpose of a hypercomplex extension of QT;this is why we mainly consider it below.
CHR ¨ ODINGER equation and its solutions H AMILTON vs. energy operator
According to classical mechanics, the H
AMILTON function of generalized coordinates x r an momenta p r of a system is equal to its entire energy: E = H ( p r , x r ) ≡ m X r p r + U ( x r ) (4)Replacing the variables by operators and their application to a state | φ i leads to the relationship ˆ E | φ i = ˆ H | φ i ≡ m X r ˆ p r + U (ˆ x r ) ! | φ i (5)between the energy and the H AMILTON operator which is nothing less than the S
CHR ¨ ODINGER equation in terms ofmatrix mechanics. Note that ˆ H and ˆ E are essentially different operators - (5) were trivial otherwise - because ˆ E depictsthe temporal behaviour of | φ i , ˆ H its spatial behaviour and the effects of a potential. Of course, they share the sameeigenfunctions | φ ( E ) i corresponding to the same eigenvalues E . For an ˆ E - (or ˆ H -)eigenstate | φ ( E ) i , the operator ˆ E canbe replaced by the value E which leads to the stationary Schr¨odinger equation ˆ H | φ ( E ) i = E | φ ( E ) i . (6)To link to wave theory, we use (3) to express the momentum operators and the energy operator in position representation: ˆ p r = i − ∂ x r = − i∂ x r (7) ˆ E = − i − ∂ t = i∂ t . (8) S CHR ¨ ODINGER equation
Substituting (7) and (8) in (5) immediately yields (S
CHR ¨ ODINGER , 1926) ˆ Hφ ( ~x, t ) = (cid:18) −∇ m + U ( ~x ) (cid:19) φ ( ~x, t ) = i ∂∂t φ ( ~x, t ) . (9)In spatial representation and using (7), equation (6) becomes ˆ Hφ = (cid:18) −∇ m + U ( ~x ) (cid:19) φ = Eφ (10)whose solutions, according to (3) have the form φ ( ~x, t ) = φ ( ~x ) · e − iEt whose stationary part φ ( ~x ) already solves (10).Unlike the time-dependent solution which contains the factor e iEt but not e − iEt , this function may be real and is to beunderstood as an interference of solutions of opposite momenta, i.e. a standing wave, e.g. describing a particle in a box. and indeed is whereas the opposite is not always possible; e.g. there is no position representation of a spin state .2 Specialrelativisticwavemechanics 3 The quantization of SRT emanates from the relativistic energy-momentum-relationship (see appendix C, (99)). Like inthe S
CHR ¨ ODINGER case, replacing physical quantities by their operators leads to a differential equation (here for a freeparticle, O
SKAR K LEIN , W
ALTER G ORDON , 1926): (ˆ p µ ˆ p µ − m ) φ = ( η µρ p µ p ρ − m ) φ = 0 (11)At one hand, this equation must always be satisfied. At the other, it fails to fully depict the behaviour of the most quantumsystems not least for being 2nd order in all derivatives . Some non-number coefficients γ µ are required to set up thefollowing 1st order equation (P AUL D IRAC , 1928) [4, 5]: ( γ µ ˆ p µ − m ) φ = 0 . (12)The γ µ must neither be real nor complex, for squaring the operator on the left side yields ( γ µ ˆ p µ − m ) φ = ( γ µ γ ρ ˆ p µ ˆ p ρ + m − mγ µ ˆ p µ ) φ = ( γ µ γ ρ ˆ p µ ˆ p ρ − m − mγ µ ˆ p µ + 2 m ) φ = ( γ µ γ ρ ˆ p µ ˆ p ρ − m ) φ − m ( γ µ ˆ p µ − m ) φ | {z } =0 = 0 ⇒ ( γ µ γ ρ ˆ p µ ˆ p ρ − m ) φ = 0 , (13)where we remind the reader of the fact that γ µ γ ρ ˆ p µ ˆ p ρ is a sum containing any pair of indices in any order. For φ ( x µ ) must also solve (11), the γ µ must both anti-commute pairwise to make mixed terms cancel out ans square to ± ˆ1 whichgeneralize the numbers ± . . Altogether, they satisfy the relationship γ µ γ ρ + γ ρ γ µ = 2 η µρ ˆ1 , (14)where η µρ (also see (98)) is the metric tensor. The spatial coefficients display the same behaviour as the imaginaryunits of H , the division ring of quaternions . Indeed, the D IRAC coefficients can be interpreted using biquaternions (i.e.quaternions over C instead of R , see 3.1.3) in a more compact way than usually. A hypercomplex algebra generalizes (often extents , though not always) the field C as an algebra and hence as a vectorspace over R . Essentially, the algebra has to be unitary i.e. contain unity and hence R itself. Using a basis where 1explicitly belongs to an element of such algebra is written as [11, 7] q = a + a i + · · · + a n i n (15)where the non-real basis elements i r , r = 1 , . . . , n are often called “imaginary units” [11] regardless of the rules how theyare multiplied. We do not adopt this term for two reasons: The first one is that in algebra, the word “unit” implies theexistence of a multiplicative inverse whereas an “imaginary unit” in the above meaning can be a zero divisor which forbidsdivision by them. The second has something to do with the term “imaginary”: At least if 1 and a non-real basis elementform a 2D subalgebra, this is easy to show containing a non-real element which squares to one of the real elements -1, 0or 1; it is such an element we wish to reserve the term “imaginary” for.However, multiplication always distributes over addition from both sides [2, 24] whereas any other property of multiplica-tion like reversibility (i.e. division), associativity or even commutativity are not constitutive. These properties are exactlywhat the differences between algebras of the same dimension essentially consist of, for a basis transformation can alterthe rules of multiplication such that it becomes at least difficult to recognize an algebra. At the other hand, different rulesof multiplication don’t automatically mean a different algebra.In general, the product of two basis elements is a linear combination of the entire basis, i.e. i r i s = n X µ =0 p rsµ i µ = p rs + p rs i + · · · + p rsn i n (16)where i := 1 . Note that this has nothing to do with the imaginary unit i introduced below. In the following, we confineour considerations to algebras which have a basis in which for any ordered pair ( r, s ) and hence any product i r i s , there isat most one nonzero coefficient p rsµ , i.e. ∀ r, s ∈ { , · · · , n }∃ µ ∈ { , · · · , n } : i r i s ∈ { , − i µ , + i µ } . (17)Of course, we are going to presume such basis as given. In this case, there are finitely many possible rules of multiplica-tion, (2 n + 3) n being an upper boundary. A 2nd order equation has more solutions than a 1st order one. For example, in an n × n matrix ring, ˆ1 means the n × n unit matrix. .1 Familiarexamples 4 Subspaces and subalgebras
A (proper) subspace
U ⊂ A is a (proper) subalgebra of A iff ∀ α, β ∈ U : αβ ∈ U ∧ βα ∈ U . (18) Ideals and zero divisors
A (proper) subalgebra J ( A is a (proper) ideal of A iff ∀ γ ∈ A , β ∈ J : βγ ∈ J ∧ γβ ∈ J . (19)An algebra is called simple iff it contains no proper ideals except of { } .Two elements α, β ∈ A \ { } are called zero divisors iff α · β = 0 . In R -algebras, zero divisors use to belong to ideals.It is obvious that α ∈ J , β ∈ J are zero divisors if J ∩ J = { } . Division by β ∈ J is always impossible:- If γ / ∈ J , the equations βξ = γ and ξβ = γ have no solution ξ ∈ A , namely if γ = 1 , i.e. there is no β − .- If γ ∈ J , the solution is ambiguous at least in general due to dim A > dim J .We will see that zero divisors can play a vital role in eigenvalue equations (see appendix C.4, esp. (108)). Beside of C itself which certainly is the most famous such algebra there is also the algebra of the dual numbers whoseimaginary unit which is often called Ω squares to zero and the (much more interesting) algebra of the split-complexnumbers whose imaginary unit which is called E or σ squares to +1; we prefer σ due to the P AULI matrices which squareto the × unit matrix. They are also called hyperbolic numbers due to the property ( a + a σ )( a − a σ ) = a − a (20)which is often called the modulus and characterizes hyperbolas in the split-complex plane just like the norm of complexnumbers a circle . It corresponds to the square of the M INKOWSKI weak norm. The algebra contains the two non-trivial(i.e. non-unity) idempotent elements
12 (1 ± σ ) . (21)These three algebras are indeed the only two-dimensional hypercomplex algebras because, for a non-real basis element i with i = a + b i , a, b ∈ R , it is easy to find an imaginary element which squares to a real number and can be normalizedif non-zero [11]: i − b i + b (cid:18) i − b (cid:19) = a + b ∈ R (22) ⇒ i − b q | a + b | + δ a, − b = Ω , a = − b σ, a > − b i, a < − b (23) Unlike the examples above, the following ones contain C as a subalgebra and hence are really an extensions of thecomplex numbers. Trying to find a reversible multiplication for 3D space vectors, W ILLIAM R OWAN H AMILTON , thoughunsuccessful in his original purpose, found the quaternions [9] in 1843 by adding a real component; due to him, thealgebra was later called H . There are 3 imaginary units; a quaternion q is hence written as q = a + a i + a i + a i , a ρ ∈ R . (24)The rules of multiplication are summarized in Table 1; as H is not commutative, the order is relevant and to be understoodas row times column [9, 10]. Like in C , every q ∈ H has a conjugate q = a − a i − a i − a i (25)which can be used to compute ℜ ( q ) = q + q , ℑ ( q ) = q − q , | q | = p qq. To speak more exactly, α is called a left and β is called a right zero divisor. In [8], such ‘numbers’ are also called pseudo-nul or roots of zero. Except split-complex numbers with modulus 0 which characterize the asymptotes of the hyperbolas and are certainly zero divisors. Usually, the imaginary units are denoted by i, j, k but these symbols will be used differently. .1 Familiarexamples 5 i i i i i i i i − i − i i i − i − i i i i − i − Table 1: Multiplication of the quaternionsNote the difference from C where, in an element a + a i , it is the (real) coefficient a which is called the imaginary part,rather than a i . For the quaternion q , ℜ ( q ) , ℑ ( q ) are also called its scalar and vector part.A right quaternion q ℑ is defined by ℜ ( q ℑ ) = 0 and formally denotable as a scalar product ~v · ~ı ( ~ı := T ( i , i , i ) ). Aproduct of two right quaternions q ℑ q ℑ is − ~v · ~v + ( ~v × ~v ) · ~ı, i.e. in some sense, quaternion multiplication unifies the scalar and the cross product. Quaternions can also used to describespatial rotations[11]. The imaginary units share so many properties with spatial dimensions that this suggests to regardspace as something essentially imaginary - just like the imaginary M INKOWSKI norms of space-like four-vectors in SRT. H is a skew field or division ring , i.e. it satisfies all field axioms except of commutativity. Any plane of H containing R is a subalgebra isomorphic to C since the imaginary units are algebraically equivalent. An overview of the features of H and other algebras is provided in appendix A.2. The (H
AMILTON -C AYLEY ) biquaternions C ⊗ H are an extension of both quaternions and the bicomplex numbers dis-cussed below. They can be perceived as an algebra over C = Span( { , i } ) with three ‘outer’ imaginary units i , i , i which anti-commute pairwise while they commute with the ‘inner’ imaginary unit i , i.e. i i r = i r i =: σ r , r = 1 , , for which individually σ r = ( i r i ) = i r i = ( − · ( −
1) = +1 . (26)Like the i r , the σ r anti-commute pairwise which, ε qrs being the totally antisymmetric L EVI -C IVITA pseudo-tensor, yields σ q σ r = i i q i r = − i i r i q = − δ qr − ε qrs i s = δ qr + ε qrs i · σ s . (27)In terms of algebraic relationships, these ‘new’ imaginary units σ r are isomorphic to the P AULI matrices and hence aptto be used in relativistic QT equations like the D
IRAC equation (see appendix C) and its non-relativistic P
AULI approach.An Overview of the rules of multiplication is shown in Table 2; as above, it is to be taken as row times column. i i i i σ σ σ i i i i σ σ σ i i − σ σ σ − i − i − i i i σ − i − i − i σ − σ i i σ − i − i − σ − i σ i i σ i − i − σ − σ − i σ σ − i − i σ − σ − i i σ σ − i − σ − i σ i − i σ σ − i σ − σ − i − i i Table 2: Multiplication of biquaternions
Inner and outer conjugate
For a complex number z = x + iy, x, y ∈ R , its conjugate is unambiguously defined,namely by ¯ z = x − iy . In principle this holds for a q ∈ H for all imaginary units are equivalent.In contrast, C ⊗ H contains different types of imaginary units. Particularly, it can be understood as an algebra over C andthus a biquaternion q = α + β i + P r =1 ( β r i r + β r +3 σ r ) , α, β µ ∈ R can also be written as q = a + P r a r i r , a µ ∈ C .Beside the ‘plain’ conjugate ˘ q = α + β i + P r =1 ( β r i r + β r +3 σ r ) , q there are hence the ‘outer’ conjugate q = a − P r a r i r and the inner conjugate q ∗ = ¯ a + P r ¯ a r i r as well[17, 25]. Additionally, these types can be combined to q † = ¯ a − P r ¯ a r i r ..2 Hypercomplexgeneralizationsofoperationsusedinwavemechanics 6 An additional hypercomplex algebra containing C is the algebra C ⊗ C of the bicomplex numbers first described in 1892by C ORRADO S EGRE who had studied the quaternions before. They can be regarded as complex numbers a + i b ∈ C , a, b ∈ C := h{ , i }i with the additional ‘inner’ imaginary unit i . Unlike their superalgebra C ⊗ H , C ⊗ C iscommutative [3, 23] and contains only one ‘outer’ imaginary unit which makes it principally interchangeable with the‘inner’ one. Altogether, the multiplication rules in canonical basis are given in Table 3. i i σ i i σi i − σ − i i i σ − − i σ σ − i − i Table 3: Multiplication of bicomplex numbers (canonical basis)In contrast to H , C ⊗ H is not a division algebra but contain h{ , σ }i as a subalgebra isomorphic to the split-complexnumbers which are known to contain zero divisors. Like the latter, it contains the non-unity idempotent elements (cid:18) ± σ (cid:19) = 1 ± σ + σ ± σ ± σ , (28)each of it belonging to a purely non-real subalgebra which is even an ideal. An overview of C ⊗ C and other algebras isgiven in appendix A.2. In the following, we are going to examine the criteria a hypercomplex-valued function must satisfy to be interpreted as awave function in the S
CHR ¨ ODINGER sense:1. Oscillations and waves must be expressible by exponential functions to formulate a wave function which solves theS
CHR ¨ ODINGER equation or/and its relativistic pendants (K
LEIN -G ORDON , D
IRAC ).2. A F
OURIER transform must be applicable bidirectionally to interchange between representations (e.g. ~x , ~p ).To describe systems which cannot be measured directly, we additionally demand a purely non-real subspace (which willturn out to be a subalgebra and even an ideal) to satisfy these both conditions. In the following, the basis elements of thedemanded ideal will generally denoted by α and β whose features will be examined. CHR ¨ ODINGER -like equationsOscillations and series expansions In C (d.h. α = 1 , β = i ), E ULER ’s formula e ipx = cos( px ) + i sin( px ) , p, x ∈ R links exponential functions to trigonometric functions and hence to oscillations which is also recognizable with help ofthe T AYLOR series, its even exponent summands forming the cosine series and its odd ones the sine series multiplied by i : e ipx = ∞ X n =0 i n ( px ) n n ! = ∞ X r =0 (cid:18) i r ( px ) r (2 r )! + i (2 r +1) ( px ) (2 r +1) (2 r + 1)! (cid:19) = ∞ X r =0 ( − r ( px ) r (2 r )! + i ∞ X r =0 ( − r ( px ) r +1 (2 r + 1)!= cos( px ) + i sin( px ) (29)In a hypercomplex algebra A and its subspaces/subalgebras, the series expansion can show in a corresponding mannerwhether an exponential function αe βpx , α, β ∈ A describes oscillations and waves. For this purpose, powers must bewell-defined which requires A and its subalgebras to be at least power associative and flexible (see appendix A.1) whichis automatically satisfied by alternative and associative algebras. We propose both power associativity and flexibility. Thepower series expansion αe βpx is αe β ( px ) = α ∞ X n =0 β n ( px ) n n ! = α ∞ X r =0 (cid:18) β r ( px ) r (2 r )! + β (2 r +1) ( px ) (2 r +1) (2 r + 1)! (cid:19) = α ∞ X r =0 β r ( px ) r (2 r )! + αβ ∞ X r =0 β r ( px ) (2 r +1) (2 r + 1)! . (30).2 Hypercomplexgeneralizationsofoperationsusedinwavemechanics 7To make the functions represented by (30) periodical, β must behave like an imaginary unit in the sense of C , i.e., theremust be γ ∈ A whose span is isomorphic to R and which satisfies β = − · γ . If so, there is also λ ∈ R with γ = λγ .This implies λ − γ =: ǫ to be idempotent , i.e. ǫ m = ǫ ∀ m ∈ N (including the possibility of ǫ = 1 ). Then, β = − λ ǫ and αe β ( px ) = αǫ ∞ X r =0 ( − r ( λ ( px )) r (2 r )! + αβλ ǫ ∞ X r =0 ( − r ( λ ( px )) (2 r +1) (2 r + 1)!= αǫ cos( λ ( px )) + α βλ ǫ sin( λ ( px )) . (31)Within the first line, we used the idempotency of ǫ to factor it out thus obtaining functions of real arguments. Forsimplicity, we assume λ = 1 . Obviously, Span( { ǫ, β } ) is a subalgebra of A which is isomorphic to C and might alsocontain α (not necessarily, as purely imaginary oscillations in H show). The role of the idempotent element
Idempotent elements like ǫ must be either 1 or zero divisors because ǫ = ǫ ⇒ ǫ · ǫ = 1 · ǫ ⇒ ( ǫ − ǫ = 0 , (32)and thus our proposal that Span( { ǫ, β } ) is a purely non-real subalgebra of A implies that A cannot be a division algebra. Oscillation and differential equations
A ‘deeper’ approach to oscillations than that via series and trigonometric func-tions are differential equations because they elementarily describe the behaviour of a system. A function f ( x ) which is todepict a harmonic oscillation with x being the phase must solve a differential equation of the form ∂ x f ( x ) = − p f ( x ) . (33)If f ( x ) = αe βpx and α, β ∈ A , ∂ x αe βpx = αβ p e βpx ! = − αp e βpx ⇒ α ( β + 1) = 0 , (34)which implies β = − if A is simple and does not contain any zero divisors. Schr¨odinger equation for free particles
The S
CHR ¨ ODINGER equation is a kind of wave equation which relates mo-mentum and (in free particle case kinetic) energy. Thus, for a momentum and energy eigenstate φ , p m φ = Eφ.
Using the ansatz φ = αe β ( px − Et ) , the first derivative with respect to t is ∂ t φ = αβ ( − E ) e β ( px − Et ) = − Eαβe β ( px − Et ) = ∓ Eβφ, if αβ = ± βα, (35)Thus β α = αβ = − α leads to β∂ t φ = ∓ Eβ φ = ± Eφ, (36)because φ contains α as a factor. The 2nd derivative with respect to x is ∂ x φ = αβ p e β ( px − Et ) = − p φ, (37)making φ be an eigenfunction of the operator − ∂ x corresponding to the eigenvalue p . Thus the S CHR ¨ ODINGER equationtakes the form − ∂ m∂x φ = ± β ∂∂t φ (38)depending on whether α and β commute or anti-commute. Oscillation and S
CHR ¨ ODINGER equation in quaternions
The quaternions have infinitely many subalgebras whichare isomorphic to C and hence allow oscillations; their basis elements are unity and an arbitrary unit right quaternion which is defined by ~ı a = a i + a i + a i with a + a + a = 1 . Since the i r anti-commute pairwise, making mixed terms cancel out, ~ı a = a i + a i + a i = ( − a + ( − a + ( − a = − , .2 Hypercomplexgeneralizationsofoperationsusedinwavemechanics 8it is isomorphic to i ∈ C . A function e ~ı a px thus depicts an oscillation which certainly holds for ~ı b e ~ı a px where ~ı b = b i + b i + b i mit b + b + b = 1 is another unit quaternion. If, additionally, ~ı a ⊥ ~ı b , i.e. P r =1 a r b r = 0 , the oscillation takes place within a purelyimaginary subspace. Such an exponential function within a purely imaginary plane may e.g. be i e i px = i · (cid:18) i px − px ) − i ( px )
3! + 1 ( px )
4! + i ( px )
5! + . . . (cid:19) = i + i px − i ( px ) − i ( px )
3! + i ( px )
4! + i ( px ) − . . . = i cos( px ) + i sin( px ) . (39)Of course, such a function also satisfies (34). According to (38) and using φ = i e i ( px − Et ) , the pairwise anti-commutativity of the imaginary unit leads to a free particle S CHR ¨ ODINGER equation − ∂ m∂x φ = − i ∂∂t φ. (40)Thus quaternions allow oscillations to be depicted by exponential functions and even a S CHR ¨ ODINGER equation to beformulated even with a purely imaginary wave function, though with the time derivative having a negative sign in contrastto the complex case.
OURIER transform
In the following, we elaborate the criteria for a F
OURIER transform to be implemented within a plane of A by denotingthe basis elements of the plane by α and β and by examinating the conditions for their multiplication rules. Starting from 1D-F
OURIER transform in C A function F ( x ) can often be written as a sum of many periodic functionsor at least as an integral over a continuum of functions G ( p ) : F ( x ) = 1 √ π Z G ( p ) e ipx d p (41)The function of amplitudes is computable via G ( p ) = 1 √ π Z F ( x ) e − ipx d x (42) Hypercomplex generalizations
In the following, the above procedure is generalized to two hypercomplex elements α and β yet not specified: F ( x ) = 1 √ π Z G ( p ) αe βpx d p (43) G ( p ) = 1 √ π Z F ( x ) αe − βpx d x (44)A concrete value of F can be extracted via D IRACS ’s delta function defined by the identity Z ∞−∞ f ( x ) δ ( x ) d x = f (0) ∀ f ( x ) (45)and hence F ( x ) = Z ∞−∞ F ( x ′ ) δ ( x − x ′ ) d x ′ . (46)Using the hypercomplexly generalized integral representation of the delta function, δ ( x − x ′ ) = 1 √ π Z ∞−∞ αe βp ( x − x ′ ) d x ′ , (47) This is actually a distribution which is a specific functional acting on functions rather than on numerical values. It can be interpreted as a functionvia the nonstandard analysis formulated by A
BRAHAM R OBINSON in 1961 which defines different nonzero infinitesimals and infinite elements, e.g. asa normalized G
AUSS function with an infinitesimal standard deviation. .3 Non-realcomplex-isomorphicsubalgebrasofthebicomplexnumbers 9this is Z ∞−∞ F ( x ′ ) δ ( x − x ′ ) d x ′ = 1 √ π Z ∞−∞ F ( x ′ ) d x ′ Z ∞−∞ αe βp ( x − x ′ ) d p = 1 √ π Z ∞−∞ F ( x ′ ) αe − βpx ′ d x ′ Z ∞−∞ αe βpx d p (48) = 1 √ π Z ∞−∞ G ( p ) αe βpx d p from which following conditions for the exponential function emanate: αe βp ( x + x ′ ) = αe βpx · αe βpx ′ (49) e ix e ix ′ = (cos x + i sin x )(cos x ′ + i sin x ′ )= cos x cos x ′ + i cos x sin x ′ + i sin x cos x ′ + i sin x sin x ′ (50) = cos( x + x ′ ) + i sin( x + x ′ ) αe βx αe βx ′ = αe β ( x + x ′ ) h (49) = α cos( x + x ′ ) + β sin( x + x ′ ) i = ( α cos x + β sin x )( α cos x ′ + β sin x ′ ) (51) = α cos x cos x ′ − α sin x sin x ′ + β sin x cos x ′ + β sin x ′ cos x = α cos x cos x ′ + β sin x sin x ′ + βα sin x cos x ′ + βα sin x ′ cos x By comparing the coefficient we obtain α = α β = − α αβ = βα = β. (52)Thus the subalgebra has to be isomorphic to C anyway, i.e. have the same rules of multiplication. For a purely imaginarysubalgebra, this means that α must be an internal identity element (and hence a zero divisor, due to (32)). Application to the quaternions As H is a division algebra , it cannot have subalgebras with internal identity elementsand so fails to satisfy our proposals for F OURIER transforming within purely imaginary subalgebras.
From (28) we already know that the bicomplex numbers contain the non-unity idempotent elements ± σ which are both‘candidates’ for k . We choose σ =: k which makes − σ = 1 − k = k . Beside these elements there are ( i − i ) =: j with (cid:18) i − i (cid:19) = i − i i + i − − σ − k and σ i − i i − i and further ( i + i ) = i − j = j with (cid:18) i + i (cid:19) = i + 2 i i + i σ −
12 = k − and − σ i + i i + i . Since j and k are linearly independent separately and with and i as well, they can be used as basis elements insteadof i , i . If we depict the canonical basis as orthogonal, the j - and k -axes are diagonal. Hence we refer to this basis { , i, j, k } shortly as a oblique basis. These multiplication rules are listed in Table 4.The bicomplex numbers thus have four C -isomorphic subalgebras, two of them being purely non-real (see Table 5). The subalgebras J , J consist of elements which are ‘ · ′ -conjugates of each other. Additionally, they are ideals with J ∩ J = { } which implies ab = 0 ∀ a ∈ J , b ∈ J e.g. / σ )(1 − σ ) = k (1 − k ) = 0 and / i − i )( i + i ) = j ( i − j ) = 0 .Below we focus on Span( { , i } ) = C and Span( { σ, i − i } ) = Span( { k, j } ) = J . The non-real elements k and k = 1 − k are inter-convertible. The angle brackets and the braces within them mean linear span and can also be denoted by
Span( { , i } ) . .4 ApplicationofthebicomplexnumbertoQT 10 i j k i j ki i − − k jj j − k − k jk k j j k Table 4: Multiplication for the bicomplex numbers represented by its oblique basisSymbolic Denotation canonical basis oblique basis C h{ , i }i h{ , i }i C h{ , i }i h{ , ( i − j ) }iJ h{ σ, i − i }i h{ k, j }iJ h{ − σ, i + i }i h{ − k, i − j }i = h{ k, j } Table 5: C -isomorphic planes in C ⊗ C In the following, we examine how linear operators known from QM act on C - and J -valued wave functions. For simplic-ity, we focus on plane waves with a certain wave vector ( b = momentum) ~p (wave number p in 1D). If we denote (3) by φ C and interpret C - where the values come from - as a subalgebra, its J -valued pendant with thesame ~p and E has the form φ J ( ~x, t ) ∝ ke j ( ~p · ~x − Et ) . (53)By the way, the latter equals ke i ( ~p · ~x − Et ) as well because k · i = k · j = j (54)like series expansion can show. For our rather elementary consideration only requires 1D, we rewrite the functions as φ C = e i ( px − Et ) (55) φ J = ke j ( px − Et ) (54) = ke i ( px − Et ) = kφ C . (56)Both functions can be interpreted as parts of an entire wave function φ = φ C + φ J . Even e jpx can be denoted by φ J asit is seen by the series expansion, namely φ J − k + 1 . CHR ¨ ODINGER equation
Again we start from standard QM. The partial wave function φ C is the eigenfunction of the momentum operator − i∂ x corresponding to the eigenvalue p : − i∂ x φ C = − i∂ x e i ( px − Et ) = − i · i | {z } =1 pe i ( px − Et ) = pφ C . (57)The operator should also apply to the entire wave function which implies that it should apply to the partial wave function φ J as well; the latter turns out to be an eigenfunction of the same operator corresponding to the same eigenvalue as well: − i∂ x φ J = − i · jpke j ( px − Et ) = − i · jpe j ( px − Et ) = pke j ( px − Et ) = pφ J = − j∂ x φ J (58)Reversely, the k -fold of the momentum operator should apply to the entire wave function and therefore to φ C ( x ) as well,and via k · ( − i∂ x ) φ C = − j∂ x φ C = − j · ipe i ( px − Et ) = − j · ipe i ( px − Et ) = pke i ( px − Et ) = kpφ C = k · − i∂ x φ C , (59)this leads to a non-real eigenvalue kp . In contrast, the application of this operator to φ J yields k · ( − i∂ x ) φ J = − j∂ x φ J = − j · jpke j ( px − Et ) = − j · jpe j ( px − Et ) = pke j ( px − Et ) = pφ J = kpφ J , (60)i.e. the eigenvalue is ambiguous inasmuch as φ J , as an eigenfunction of the operator, can be interpreted as correspondingboth to kp and to p . A physical interpretation of this result will be considered in future examinations. However, the onlyway to obtain an unambiguously non-real eigenvalue is to apply an J -valued operator on an at least partly C -valued wave .4 ApplicationofthebicomplexnumbertoQT 11 function .As an eigenfunction of the momentum operator corresponding to the eigenvalue p , both φ C and φ J obviously solve theS CHR ¨ ODINGER equation (9), e.g. for U = 0 : ˆ H C φ C = − ∂ x m φ C = − i p m φ C = p m φ C = i ∂∂t φ C = i · ( − i ) Eφ C = Eφ C (61) ˆ H C φ J = − ∂ x m φ J = − j p m φ J = kp m φ J = p m φ J = i ∂∂t φ J = i · ( − j ) Eφ J = kEφ J = Eφ J (62)Applying the k -fold of the S CHR ¨ ODINGER equation to functions yields ˆ H J φ J = − k ∂ x m φ J = − kj p m φ J = k p m φ J = j ∂∂t φ J = j · ( − j ) Eφ J = kEφ J = Eφ J (63) ˆ H J φ C = − k ∂ x m φ C = − ki p m φ C = k p m φ C = j ∂∂t φ C = j · ( − i ) Eφ C = kEφ C . (64)This shows that the S CHR ¨ ODINGER equation in both the C and the J form (i.e. with or without k which can never be gotout if once in because J is an ideal) applies to φ J , obtaining the same ambiguity as with the momentum operators. Conclusion:
For kφ J and φ J are indistinguishable, the partial wave function φ J leads to eigenvalues which can beinterpreted as k -valued but also as real as well. It is distinguishable only whether the normal operator or their k -fold areapplied to the C -valued function. For physical interpretation, this suggests to regard the operators, rather than the wavefunctions, as the extension of QT which is made even more plausible as far as in the description of photons[12], the carrierof the actual physical quantities like e.g. the electric field intensity is not the wave function but the operators. OURIER transform
Just like φ C , φ J should have a momentum and energy representation which is obtained by F OURIER transform accordingto (44). In 1D which is clearly sufficient for showing it in principal, this is φ J ( p ) = 1 √ π Z ∞−∞ φ J ( x ) ke − jp ′ x d x = 1 √ π Z ∞−∞ ke j ( p − p ′ ) x − Et ) d x = kδ ( p − p ′ ) e jEt . (65)The delta function is obtained by, roughly speaking, phase factors cancelling out within the infinitely narrow p -range p ′ = p , leaving the integrand constant and thus the integral infinite. This does not happen if one tries to apply the C F OURIER transform to φ J : √ π Z ∞−∞ φ J ( x ) ke − ip ′ x d x = 1 √ π Z ∞−∞ ke ( jp − ip ′ ) x − jEt d x (66)Here the integrand stays periodic for p ′ = p , leaving the integral bounded. This holds for applying the J F OURIER transform to φ C . A physical interpretation of this will be given in future examinations. In QT, the absolute square φ ( x ) φ C ( x ) of a wave function of an observable x is interpreted as a probability densityfor measuring a certain value of x , the wave function being complex-valued and hence the conjugation unambiguouslydefined . As we have seen, in C ⊗ H different types of conjugates can be defined thus and in C ⊗ C as well. Beside a“plain” conjugate which maps any imaginary component to its negative, there is an ‘outer’ one which does so with i andthus with σ and an ‘inner’ which maps i and σ to their negatives. Both map an element from J to one from J whoseproduct with the former is always zero, hence yielding φ J φ J ≡ .There is also a combined or double conjugate of q ∈ C ⊗ C which is defined by q † := q ∗ = q ∗ . For i † = − i , i † = − i i , σ † = ( − i )( − i ) = σ and thus k † = k, j † = − j , the conjugate of a J -valued function being like in C just with k in theplace of 1.Of course, the product q † q , still being non-real as it contains the factor k , may not be called absolute square; according tothe wording for split-complex numbers, we call it the “modulus”. The modulus of a momentum operator eigenfunction isspatially constant; its non-real value is to point up that it is not a probability density which were measurable in principal: ke jpx ke − jpx = ( k cos( px ) + j sin( px ))( k cos( px ) − j sin( px ))= k cos ( px ) − j sin ( px )= k (cid:0) cos ( px ) + sin ( px )) (cid:1) = k. (67) Or in terms of nonstandard analysis where infinite and infinitesimal quantities are well-defined. Or φ ∗ C ( x ) φ C ( x ) like usual in physics This does not allow with equalize “imaginary” to “not measurable” with “real” to “measurable”! The real part of a usual QT wave function is nomore measurable than the imaginary part. Reversely, SRT suggests to equalize “imaginary” with “space like” when M
INKOWSKI norms are considered. expectation value or some pendant of it, respectively. As theexpectation value of the operator − i∂ x in the state φ C is naturally h φ C | − i∂ x | φ C i = e − i ( px − Et ) · − i∂ x e i ( px − Et ) = e − i ( px − Et ) · − i · i · p · e i ( px − Et ) = p (68) h φ C | i∂ t | φ C i = e − i ( px − Et ) · i∂ t e i ( px − Et ) = e − i ( px − Et ) · i · − i · E · e i ( px − Et ) = E, (69)and taking φ † as the conjugate, the expectation value of the same operator in the state φ J is h φ J | − i∂ x | φ J i = ke − j ( px − Et ) · − i∂ x ke j ( px − Et ) = kp = e − i ( px − Et ) · − j∂ x e i ( px − Et ) = h φ C | − j∂ x | φ C i = ke − j ( px − Et ) · − j∂ x ke j ( px − Et ) = h φ J | − j∂ x | φ J i . (70) h φ J | i∂ t | φ J i = ke − j ( px − Et ) · i∂ t ke j ( px − Et ) = kE = e − i ( px − Et ) · j∂ t e i ( px − Et ) = h φ C | j∂ t | φ C i = ke − j ( px − Et ) · j∂ t ke j ( px − Et ) = h φ J | j∂t | φ J i (71)and thus is k -valued wherever the wave function or the operator is J -valued, even where the eigenvalue is ambiguous.Note that ke i ( px − Et ) = ke j ( px − Et ) and hence kφ C = φ J . Initially we sketched QT in its fundamentals and saw that, in Newtonian approximation, its formulation requires complexnumbers (or something isomorphic to it).Further we saw that a correct and complete relativistic QT (especially the D
IRAC equation) requires even more, i.e. ahigher dimensioned and non-commutative hypercomplex algebra for its coefficients.Before we went into details, we first described the general properties of hypercomplex algebras. Then we consideredsome examples of low dimension, some of which being extensions rather than generalizations of C . Beside the divisionring or skew field H of the quaternions which is by far the best known hypercomplex algebra we became acquainted withan extension of H , namely the algebra C ⊗ H of the (H AMILTON -C AYLEY ) biquaternions which soon turned out to beapt to formulate the D
IRAC equation though some difficulties of interpretation arose which are to be concerned about infuture examinations. Beside the ‘plain’ conjugate which means to negate all imaginary components, the biquaternionsprovide different kinds of conjugates which we called the ‘inner’ and the ‘outer’ one, and their combination as well.Subsequently we considered C ⊗ C , a subalgebra of the bicomplex numbers which, in contrast to H , is commutativeand, like the biquaternions, contains zero divisors and hence elements division by which is impossible and some of whichbeing idempotent that later turned out to be important.Our main issue was an extension for QT with a hypercomplex algebra which at least contains one purely non-real sub-space S such that S -valued QT should be performable in the same manner as in normal complex values. This impliesthat S -valued exponential functions should describe oscillations and waves and so the formulation and solution of aS CHR ¨ ODINGER equation should be able as well, which still holds for the quaternions.Furthermore, it implies the possibility of F
OURIER transforms to change the basis from position to momentum represen-tation and vice versa . Such a purely non-real sub space turned out to have to be a sub algebra isomorphic to C . From thisfollows the existence of an internal identity element which must be idempotent and, for being nun-unity, also a zero divi-sor, thus making the subalgebra S be (or belong to) a proper ideal hence denoted by J . This excludes division algebrasand therefore H .Last we found the bicomplex numbers to satisfy our postulates because they contain two idempotent elements k and k and j, j with j = − k, j = − k spanning the ideals J := Span( { k, j } ) and J := Span( { k, j } ) . Additionally, , i, j, k span the entire algebra, and we use them as the new basis.In the end, we introduced two partial wave functions φ C = e ipx and φ J = ke jpx and applied the original S CHR ¨ ODINGER equation and its J -valued version to both. The eigenvalue obtained by the application of the J -valued version to φ J turned out to be ambiguous insofar as it can be interpreted as k -valued but as real-valued as well. Last we used thecombined conjugate defined above to assign a nonzero modulus to φ J and to compute expectation values for the state φ J which, in contrast to the eigenvalues, all are unambiguous.Future examinations will have to physically interpret the J -valued partial wave functions and the C ⊗ C -valued entirewave function according to our results here. Acknowledgement
We give thanks to Hans R. Moser for the inspiring debates and some critical advice which helpedus to develop this paper.EFERENCES 13
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A Superordinate properties of hypercomplex algebras
A.1 general properties
Distributivity is both related to addition and multiplication at once inasmuch as the latter distributes over the former.All other properties are related to both individually . In the following, we consider these properties of the multiplicationbecause in algebras, addition is always associative, commutative and reversible.
Distributivity
Distributivity, which means a ( b + c ) = ab + ac ( b + c ) a = ba + ca (72)is a basic proposition for any hypercomplex algebra. Associativity and its dilutions
An algebra A is called associative if ( ab ) c = a ( bc ) ∀ a, b, c ∈ A . (73)Examples are, of course, R , C and H and all n × n matrix rings as well. Reversely, associative hypercomplex algebrashave a matrix representations [2], unity being represented by n × n unit matrices. Eventual zero divisors then show up assingular matrices. Another formulation for associativity is that the associator [ a, b, c ] =: ( ab ) c − a ( bc ) vanishes. A is called alternative if ( aa ) b = a ( ab ) ∀ a, b ∈ A . (74)One example is the algebra O of the octonions but all associative algebras are alternative as well. The name is due to thefact that the associator is alternating, i.e. [ a, b, c ] = − [ a, c, b ] and so on [2, 19]. A is called flexible if ( ab ) a = a ( ba ) ∀ a, b ∈ A (75)and power-associative if a m + n = ( a m )( a n ) ∀ a ∈ A , m, n ∈ N . (76)One example is the algebra S of the sedenions but all alternative algebras are both flexible and power-associative as well. Commutativity and anti-commutativity A is called commutative or rather anti-commutative if ab = ± ba ∀ a, b ∈ A ; (77)this is immediately visible in the multiplication table since this is symmetric or anti-symmetric to the main diagonal.However, strict anti-commutativity does not exist in hyperkomplex algebras because they contain the real numbers whichcommute with any other element. Nevertheless, there will be certain anti-commuting elements unless A is commutative. General reversibility of multiplication A is called a division algebra if z z = z and zz = z (78)have a unique solution z for all z , z ∈ A . If z is a zero divisor and belongs to an ideal I , respectively, there is nosolution for z / ∈ I and many, often even a whole continuum of solutions for z ∈ I . A.2 Properties of the algebras examined in this paper
For the algebras explicitly mentioned and examined in this paper, we summarize their properties, i.e. the properties of themultiplication, in a table: Biquaternions and bicomplex numbers do not form division algebras, as, particularly well seenin the so-called oblique basis (table 4) where zero divisors are basis elements denoted here as k and j . The columns androws for j and k neither contain nor i but each two incidents of j and k instead.6name symbol distributive associative commutative reversiblecomplex numbers C yes yes yes yesdual numbers - yes yes yes nosplit-complex numbers - yes yes yes noquaternions H yes yes no yesbiquaternions C ⊗ H yes yes no nobicomplex numbers C ⊗ C yes yes yes noTable 6: Properties of several B Formalism of QT
B.1 Summary of the most important basic concepts H ILBERT spaces and quantum states
Matrix mechanics generalizes the analytical geometry of the familiar 3D spacewhich is a special case of vector spaces over R or C with named after D AVID H ILBERT : It has a scalar product andtherefore the euclidean norm and additionally is complete , i.e. all C
AUCHY sequences converge within the space. These properties are common to any H ILBERT space.In QT, a quantum state is represented by a vector from H which, according to P AUL D IRAC , is denoted by | φ i . Anycomplex multiple z | φ i , z ∈ C represents the same state of a particle or a system, thus the state itself which | φ i representscan be identified with Span( | φ i ) ⊂ H which is actually a whole 1D subspace.H ILBERT spaces can have very different dimension including infinite and even uncountably infinite. An example for aH
ILBERT space of such dimension is the function space L ( R ) of which links to wave mechanics : The wave function φ ( ~x, t ) is straightly a specific (here: position) representation of the quantum state | φ i . Position space, according to itsproperties, is clearly itself a H ILBERT space but as far as this H
ILBERT space of spatial functions is concerned, it is just akind of index set.
Combination of several H
ILBERT spaces
Tensor product H = H (1) ⊗ · · · ⊗ H ( n ) of n H ILBERT spaces is itself aH
ILBERT spaces, its elements being | φ i = | φ i · · · | φ i n . Note that the H r can be completely different. There are manycases where such the Combination is required to provide a complete description of particles especially if they have a spin . For example, for a spin particle such as the electron, its spin H ILBERT space being H = C . Thus a completedescription of such a particle requires the tensor product L ( R ) ⊗ C its elements being the solutions of W OLFGANG P AULI ’s equation.
Dual space and scalar product
A quantum state | φ i ∈ H corresponds to a vector h φ | of H ∗ , the dual space of H whichis actually a linear map H → K , namely the map of an arbitrary vector | ψ i to its scalar product with | φ i which is thusdenoted by h φ | ψ i . In general, K = C . Perhaps according to duality, the complex conjugate is often denoted as z ∗ insteadof z in QT. Normalization and orthonormal basis
As a H
ILBERT space, H consists of elements which have a norm by it canbe divided to normalize it. Hence, L ( R ) is defined by consisting of square integrable functions φ ( ~x ) for which R φ ∗ φ d x < ∞ . The function | φ i and φ ( ~x, t ) , respectively, is called normalized if h φ | φ i = Z { ~x } φ ∗ ( ~x, t = const . ) φ ( ~x, t = const . ) d x = 1 . (79)An orthonormal basis (ONB) or complete orthonormal system (CONS) is a basis {| r i} (where r belongs to an index setwhich may be continuous) of H with h r | s i = δ rs = ( , r = s , r = s (80)It is a somewhat annoying that ansatz (3) itself lacks a norm and thus does not actually belong to L ( R ) . However,strictly periodical functions (i.e. such with sharply defined ~p ) are something idealized.Multiplication by a, extremely flat-angle normalized function leads to a square-integrable wave function whose progressis hardly discernible from (3) over a wide range. In the following, the functions are to be assumed as normalized. Unlike Q because there are rational C AUCHY sequences with an irrational limit. Preferably a G
AUSSIAN since it is its own F
OURIER transform. .2 ExamplesofnotationinQT 17
Operators
The concept of a matrix is generalized in H by that of a linear operator ˆ A . With reference to a certain CONS | r i , ˆ A has a matrix representation h r | ˆ A | s i where r, s are indices which are continuous if H is a function space. If ˆ A represents an observable A , it is Hermitian, i.e. h s | ˆ A | r i = h r | ˆ A | s i ∗ which implies h r | ˆ A | r i ∈ R ; this matrix element iscalled the expectation value of ˆ A in the state | r i . Eigenvalues and eigenvectors, measurements
A quantum state | v i for which ˆ A | v i = a v | v i is called an eigenstate of ˆ A corresponding to the eigenvalue a v ∈ A = { a } and represents a quantum state where measurements of A yieldthe value a v without emphprincipal deviations. Of course, a v is the expectation value of ˆ A in the state | v i as well, and h v | ˆ A | v i = h v | a v | v i = a v h v | v i = a v . Expansion in eigenstates, F
OURIER transform
Anything which can be measured are eigenvalues of Hermitian oper-ators like e.g. ˆ A which holds for the case that | φ i not an eigenstate of ˆ A because it can be EXPANDED in eigenstates of ˆ A which generalizes linear combination: | φ i = X a ∈ A z ( a ) | a i bzw. | φ i = Z A z ( a ) | a i d a (81)There z ( a ) is the complex probability amplitude and z ∗ ( a ) z ( a ) ≡ | z ( a ) | is the probability or probability density of ameasurement of a in the state | φ i . An example for the expansion of a wave function φ ( ~x, t ) in functions of the type (3)which is actually the F OURIER transform φ ( ~p, t ) = F ( φ ( ~x, t )) = (2 π ) − Z { ~x } φ ( ~x, t ) e − i~p · ~x d x., (82)where φ ( ~p, t ) provides the coefficients which quantify the ratio of the momentum eigenfunction for any ~p , i.e., | φ ( ~p, t ) | is the probability density for a certain momentum measurement. Reversely, they can be used to re-compose the functionby the inverse transform φ ( ~x, t ) = F ( φ ( ~p, t )) = (2 π ) − Z { ~p } φ ( ~p, t ) e i~p · ~x d p. (83)The fact that functions that are F OURIER transforms of each other are apt to be taken as momentum and position repre-sentation of the same quantum state is due to M
ARC A NTOINE P ARSEVAL ’s theorem which says Z | φ ( ~x, t ) | d x = Z | φ ( ~p, t ) | d p. (84) Uncertainty relation
Standard deviations of such functions are reciprocal, i.e. the F
OURIER transform of a functionwith a flat progression is practically zero outside of an extremely small neighbourhood of 0 but with huge values inside,being a finite approach of D
IRAC ’s delta function . The product of these standard deviations never falls below ~ / (inconventional units), only reaching it in the case of G AUSSIANS which are a fixed point of the F
OURIER transform.This relation generally applies o two observables whose operators ˆ A, ˆ B have a fixed commutator [ ˆ A, ˆ B ] and hence noeigenstates in common (H EISENBERG , 1925). If the commutator itself is an operator, there may be common eigenstatesas this is the case for the components of an angular momentum, namely if | ~L | = 0 . B.2 Examples of notation in QT D IRAC ’s bra-ket notation allows to denote quantum states in a very abstract and general manner which contains extremeexamples like a two basis state space at one hand or a space of position wave functions containing an entire continuum ofbasis states at the other. We concretize the notation for both extreme cases.
B.2.1 Two basis state system
In this case and in matrix notation, h φ | = (cid:0) c ∗ φ, c ∗ φ, (cid:1) , | ψ i = (cid:18) c ψ, c ψ, (cid:19) ⇒ h φ | ψ i = (cid:0) c ∗ φ, c ∗ φ, (cid:1) (cid:18) c ψ, c ψ, (cid:19) = n X r =1 c ∗ φ,r c ψ,r . (85)In such a H ILBERT space and with respect to some given standard basis then written as (cid:26)(cid:18) (cid:19) , (cid:18) (cid:19)(cid:27) , .2 ExamplesofnotationinQT 18 ˆ A is a × matrix ( a rs ) , r, s = 1 , and h φ | ˆ A | ψ i = (cid:0) c ∗ φ, c ∗ φ, (cid:1) (cid:18) a a a a (cid:19) (cid:18) c ψ, c ψ, (cid:19) = n X r =1 n X s =1 c ∗ φ,r a rs c ψ,s . (86)If | φ i and | ψ i form a basis of H as well, ˆ A is represented by matrix elements (cid:18) h φ | ˆ A | φ i h φ | ˆ A | ψ ih ψ | ˆ A | φ i h ψ | ˆ A | ψ i (cid:19) , (87)with respect to this basis, the diagonal elements being the expectation values of ˆ A in the states Span( | φ i ) and Span( | ψ i ) . Spin system as an example
Eigenvalues of spin direction are always projections of the spin to a given axis. The z axis traditionally is the rotation axis in 3D space like it is easily seen by means of the definition of spherical coordinates.Hence it is conventional to take the orientation relatively to the z axis as the standard basis. Eigenstates in other directionsmay be expanded in z eigenstates, of course; for example, the y eigenstates are denoted by √ | + i ± i |−i ) = 1 √ (cid:18) ± i (cid:19) (88)according to convention. They are the eigenstate of the P AULI matrix σ = σ y : (cid:18) − ii (cid:19) (cid:18) ± i (cid:19) = (cid:18) ± i (cid:19) . (89)In the ‘+’ case, the vector corresponds with the eigenvalue 1, in the ‘-’ case with the eigenvalue -1 (the scale factor canbe omitted in eigenvalue equations). These Eigenvalues are certainly the expectation values of the operator σ y in theeigenstates (88) as well. The non-diagonal elements provide 0 because both eigenstates are orthogonal. Thus the operatoris (cid:18) ( h + | + i h−| ) σ y ( | + i + i |−i ) ( h + | + i h−| ) σ y ( | + i − i |−i )( h + | − i h−| ) σ y ( | + i + i |−i ) ( h + | − i h−| ) σ y ( | + i − i |−i ) (cid:19) = (cid:18) − (cid:19) (90)with respect to the basis of its own eigenstates, exactly like the operator σ or σ z in the standard basis. B.2.2 Position wave function If H = L ( R ( ~x )) , the vectors are functions and sums become integrals: h φ | ~x i = φ ∗ ( ~x, t ) , h ~x | ψ i = ψ ( ~x, t ) ⇒ h φ | ψ i = Z { ~x } φ ∗ ( ~x, t ) ψ ( ~x, t ) d x (91)In this case and with respect to | φ i , | ψ i , the matrix element is h φ | ˆ A | ψ i = Z { ~x } φ ∗ ( ~x, t ) ˆ A ψ ( ~x, t ) d x (92)or, more generally h φ | ˆ A | ψ i = Z { ~x } Z { ~x ′ } φ ∗ ( ~x, t ) h ~x | ˆ A | ~x ′ i ψ ( ~x ′ , t ) d x d x ′ . (93)For | ψ i = | φ i , this is the expectation value. If h ~x | v i = φ v ( ~x, t ) is an eigenstate of ˆ A corresponding to the eigenvalue a v , h v | ˆ A | v i = Z { ~x } φ ∗ v ( ~x, t ) ˆ A φ v ( ~x, t ) d x = Z { ~x } φ ∗ v ( ~x, t ) a v φ v ( ~x, t ) d x = a v Z { ~x } φ ∗ v ( ~x, t ) φ v ( ~x, t ) d x = a v , (94)exactly as it should be since the expectation value must equal the eigenvalue because it is exact in this case.9 C Special Relativity and its quantization
C.1 Relativity principle and Special Relativity
One of the basic principles of classical mechanics is the relativity principle (RP) first discovered by G
ALILEO G ALILEI .It means that within two coordinate systems K and K ′ relatively moving in x direction the laws of mechanics are thesame, or, more formally speaking, they are invariant under G ALILEI transform which can be denoted as a matrix-vectorequation (cid:18) t ′ x ′ (cid:19) = (cid:18) − v (cid:19) (cid:18) tx (cid:19) , (95)where x is the only spatial dimension regarded here and t and x are combined to a vector which in full SR framework iscalled a four-vector.However, J AMES C LERK M AXWELL ’s basic equations of electrodynamics are not G
ALILEI invariant and neither are theelectromagnetic wave equations derived from them. This lead to the hypothesis of a luminiferous aether which transmitslight at a speed now known as c . This aether was thought to be at absolute rest. Within a moving frame - like earth’s - thespeed of light would hence vary with direction which should be measurable e.g. by interferometry. Suitable experiments,however, did not yield any deviation from RP. To explain this, H ENDRIK A NTOON L ORENTZ modified (95) step by step,finally obtaining the L
ORENTZ transform (cid:18) t ′ x ′ (cid:19) = γ (cid:18) − vc − v (cid:19) (cid:18) tx (cid:19) γ := 1 p − ( vc ) , (96)where γ is called the L ORENTZ factor. Replacing t → ct makes (96) more symmetric, yielding (cid:18) ct ′ x ′ (cid:19) = γ (cid:18) − vc − vc (cid:19) (cid:18) ctx (cid:19) , symbolically writing ⇒ x ′ = Λ( ~v ) ⇒ x . (97)The electromagnetic wave equation and hence c is invariant under L ORENTZ transform [13] and so are the M
AXWELL equations.
Especially, they satisfy
RP because unlike W
OLDEMAR V OIGT ’s 1887 transforms which also leave c invariant,L ORENTZ transforms form a group from which follows that the inverse of a L
ORENTZ transform is also a L
ORENTZ transform corresponding to the opposite velocity - symbolically speaking, Λ − ( ~v ) = Λ( − ~v ) . In 1905, A LBERT E INSTEIN based his theory of Special Relativity (SR)[6] on them and also predicted the rest energy E = mc which reverselyprovides an energy E with the mass m E = Ec − . The universal constant c is actually an artefact of the measuringsystem inasmuch as spatial and temporal distances are measured in different units. C.2 Covariant form and four-vectors
In the framework of the so-called covariant formulation of SR which was later to facilitate the coordinate-independentformulation of General Relativity (GR), ct or t is a coordinate denoted by x or x which is the same for index 0.Altogether, x µ = T ( t, x, y, z ) is called a contravariant four-vector whereas x µ = T ( t, − x, − y, − z ) is the correspondingcovariant four-vector. Both are converted into each other with help of the metric tensor η µσ = η µσ = diag { , − , − , − } (98)via x µ = η µρ x ρ and x µ = η µρ x ρ , respectively. For two four-vectors x µ , x ′ µ , a L ORENTZ invariant (weak) scalar product x µ x ′ µ = η µρ x µ x ′ ρ is defined. It is called weak or also improper because it lacks positive definiteness which is constitutivefor proper scalar products. It induces an improper or weak norm k x µ k = √ x µ x µ first mentioned by and named afterE INSTEIN ’s teacher H
ERMANN M INKOWSKI .[14, 15]
C.3 Relativistic energy momentum relation and four-momentum
The pendant of x µ in momentum space is the four-momentum p µ = T ( E, − p x , − p y , − p z ) while the pendant of x µ is p µ = T ( E, p x , p y , p z ) ; the concept of the four-momentum is justified by the energy-momentum-relationship E − ~p = p µ p µ = m , (99)i.e. mass or rest energy is (or at least os proportional to) the absolute value of the four-momentum. F RIEDRICH H ASEN ¨ OHRL had already computed a mass for cavity radiation in 1904, so the equivalence of energy and mass was new only in its general form. Note that if horizontal distances were measured in meters whereas vertical where measured in feet, this would lead to a “universal constant” κ = 0 , ft/m. .4 QuantizationofSR 20 C.4 Quantization of SR
Since a point in time t = const . is not well-defined in SR, normalization (79) of a wave function for t = const . is replacedby a continuity equation which is to emanate from the basic equation like the following. The K
LEIN G ORDON equation
Even before S
CHR ¨ ODINGER set up the non-relativistic equation named after him, hereplaced the physical quantities in (99)) by operators to set up the following differential equation (also see (11)): ˆ p µ ˆ p µ φ = −∇ µ ∇ µ φ = − (cid:3) φ := (cid:0) − ∂ t + ∇ (cid:1) φ = m φ (100)It is 2nd order in all derivations and hence there are real solutions, even time-dependent ones. These are solutions witha negative E and were thus regarded as physically impossible for a long time and rejected by S CHR ¨ ODINGER . However,it was examined further by O
SKAR K LEIN and W
ALTER G ORDON after whom it is now named (abbr.: KGE). Scrutinyreveals that even the solutions with a negative E represent a positive energy. They are the antiparticle solutions. (100)leads to the continuity equation ∇ µ ( φ ∗ ∇ µ φ − φ ∇ µ φ ∗ ) = ∇ µ ˜ µ = ∂ t ˜ ̺ + ∇ · ˜ ~ = 0 (101)which says that the four-current is a zero-divergence field. Its time component ˜ ≡ ˜ ̺ , however, is not positive definiteand hence cannot be interpreted as a probability density. Of course, it neither induces preservation of particle number.Therefore ˜ ̺ is best interpreted as a charge density or at least as a “charge probability density”. Real solutions stand forelectrically neutral K LEIN G ORDON fields for which the terms in (101) vanish individually. Neutral particles completelydescribed by KGE can hence both generated and annihilated without any violation of the equation. They are their ownantiparticles like the photon. However, the latter is a quantum of a tensor field, namely of the electromagnetic one and canthus only incompletely described by KGE.
The D
IRAC equation
In 1928, P
AUL D IRAC came up with the idea of formulating a 1st order differential equation as anansatz with initially unknown coefficients for later analysis of their required features. [4, 5] In covariant form and naturalunits, it is denoted by γ ρ ˆ p ρ φ = mφ. (102)He postulated that any function φ which satisfies (102) must satisfy (100) as well. This leads to the following commutationor rather anti-commutation relations (14). From (102), the continuity equation ∇ µ ( ¯ φγ µ φ ) = ∇ µ ( φ † γ γ µ φ ) = ∇ µ ˜ j µ = ∂ t ( φ † φ ) | {z } ˜ ̺ + ∇ ( φ † ~αφ ) | {z } ˜ ~ = 0 (103)can be derived. The expression φ † φ =: ˜ ̺ , the temporal component of the four-current, is positive definite and can hencebe interpreted as a probability density which enables (103) to express preservation of particle number . This makes theD IRAC equation apt to describe matter.Using the biquaternionic (also see 3.1.3) imaginary units σ r usually written as complex × matrices, the D IRAC coefficients may be written more concretely as γ = (cid:18) − (cid:19) , γ r = (cid:18) σ r − σ r (cid:19) . (104)Furthermore, some other coefficients γ = β, α r = γ γ r can be used to bring the equation into a S CHR ¨ ODINGER formi.e. to solve it for the temporal derivative which facilitates the computation of the non-relativistic approach. For a particlein an electromagnetic field and using ( σ , σ , σ ) =: ~σ and the kinetic momentum ˆ ~p − q ~A =: ˆ ~π , the equation hence takesthe form i∂∂t (cid:18) φ + φ − (cid:19) = ( βm + ˆ1 qA + ~α · ˆ1ˆ ~π ) (cid:18) φ + φ − (cid:19) = m + qA ~σ · ˆ ~π~σ · ˆ ~π − m + qA ! (cid:18) φ + φ − (cid:19) . (105)In the limit of vanishing velocities and fields, this becomes i∂∂t (cid:18) φ + φ − (cid:19) = (cid:18) m − m (cid:19) (cid:18) φ + φ − (cid:19) . (106)The case E = + m implies φ J = 0 , the case E = − m implies φ + = 0 ; hence φ + represents matter and φ − antimatter[1, 18, 26]. In cases of high energies both occur, thus impeding a one-particle-description like in S CHR ¨ ODINGER case.For each case of E = ± m , the P AULI equation can be derived which in ‘+’ case can be written as i ∂∂t ξ (ˆ ~p − q ~A ) − q~σ · ( ∇ × ~A )2 m + qA ! ξ, ξ = φ C e − imt (107)1The σ r are the components of the spin operator. If they are taken as matrices, the spin states are denoted by C vectors. Ifthe operator components are written as biquaternions instead, the states must be biquaternions as well: σ r (1 ± σ r ) = σ r ± ± (1 ± σ r ) (108)This means that (1 ± σ r ) , as a state, is an ‘eigen-biquaternion’ of σ r corresponding to the eigenvalue ± . This impliesthat the state biquaternion is a zero divisor, its plane, outer or inner conjugate as a ‘zero divisor partner’. Obstacles of the interpretation
Representing both operators and states by elements of the same algebra, i.e. thebiquaternions blurs the difference between them. A further difficulty is the necessity to define scalar products and normsfor zero divisors for which the multiplication with its conjugate is not useful. Furthermore, spin eigenstates of an operatorfor a certain direction should be able expanded in eigenstates of an operator for another. Using matrix-vector-notation,this is obtained without force whereas biquaternions resist because the σ r are linearly independent. These problems mayhave impeded that biquaternion formulation could have successfully competed with matrix-vector-formulation. D Prefactors in differential operators in a H
ILBERT space over the ideal
In the following, partial derivatives of functions of type (3) and (53) are provided with different pre-factors from J arelisted (underlined results also apply for φ J = kφ C ): + j∂ x φ C = + j · ipe i ( px − Et ) = − kpe i ( px − Et ) = − kpφ C = − kpφ J = − pφ J (109) + j∂ x φ J = + j · jpke j ( px − Et ) = j pe j ( px − Et ) = − kpe j ( px − Et ) = − kpφ J = − pφ J (110) − k∂ x φ C = − k · ipe i ( px − Et ) = − jpe i ( px − Et ) = − jpφ C = − jkpφ C = − jpφ J (111) − k∂ x φ J = − k · jpke j ( px − Et ) = − jpe j ( px − Et ) = − jpφ C = − jkpφ C = − jpφ J = − ipφ J (112) + k∂ x φ C = + k · ipe i ( px − Et ) = jpe i ( px − Et ) = jpφ C = jkpφ C = jpφ J (113) + k∂ x φ J = + k · jpke j ( px − Et ) = jpe j ( px − Et ) = jpφ C = jkpφ C = jpφ J = ipφ J . (114)The + j -valued operators yield a negative sign which were correct in case of the temporal derivative. The k -valuedoperators yield purely imaginary eigenvalues and therefore are anti-Hermitian regardless of their sign. If QT in J is towork equivalently to QT in C , eigenvalues should be pseudo-real, i.e. kk