On the quantum theory of rotating electrons

Transcription

1 Zur Quantentheorie des rotierenden Elektrons Zeit. f. Phys. 8 (98) On the quantum theory of rotating electrons By Friedrich Möglich in Berlin-Lichterfelde. (Received on April 98.) Translated by D. H. Delphenich In connection with a paper of Dirac those transformation matrices will be determined that take the place of Lorentz transformations for electron spin and the eigenfunctions. Furthermore an invariant will be given from which by variation of the Dirac eigenfunctions and the electromagnetic potential the equations for the matter waves and the electromagnetic field can be obtained. In connection with that one gets an expression for the four-vector of the particle current. In an amazing recently-appearing paper P. A. M. Dirac ( * ) made relativistic quantum mechanics more powerful. The following work is connected directly with the aforementioned Dirac paper and shall moreover represent nothing but some formal extensions of the Dirac argument. Above all those transformation shall be considered here that allow one and the same evolution for the eigenfunctions as observed from different reference systems when they are converted into each other. With their help the vector properties of certain bilinear forms that are defined by the wave functions can also be verified. One further finds a Lagrange function L from which one can not only obtain the Dirac equations by variation but also the Maxwell equations. With the help of the transformations of the wave functions the proof of the invariance of L is very simple. On the contrary a series of questions that are connected with the energy-impulse tensor for matter waves will not be discussed. It may be computed very easily from the aforementioned function L and is in contrast to the tensor that was derived by Schrödinger ( ** ) no longer symmetric. We hope to return to this very singular behavior shortly. I. The spatial rotation and its associated unitary transformation. According to Dirac in the absence of an external electromagnetic field the differential equations for the rotating electron read: h e h e η + Φ ψ + η + Φ ψ π i x c π i x c imc ψ. () ( * ) P. A. M. Dirac Proc. Roy. Soc. (A) 7 (98) 6. ( ** ) E. Schrödinger Ann. d. Phys. () 8 (97) 65.

2 Möglich On the quantum theory of rotating electrons. Here: x c t Φ V Φ A Φ A Φ A. The quantities η α are the spin matrices; we (following Dirac) would like to given them the following appearance ( * ): i i i i η η + i i + i i i i η η. i i () The product (η α ψ) has the following sense: (η α ψ) ( η α ) k ψ k. k If we now subect equation () to a transformation that corresponds to a spatial rotation then / x and Φ remain unchanged while the operators / x k transform in exactly the same way as the spatial components of Φ k ; i.e. like three-dimensional vector components. Let α ik be the coefficients of the transformation matrix so equation () (under the assumption that the Φ k which is inessential) goes to: where: h ψ ψ + π i x x mc ψ () α α η α (α ) and the x k are the coordinates in the new system (so one observes that x x and η ). Dirac has proved that the quantities α emerge from the η α by a transformation of the form S η α S without however exhibiting that transformation. In order to do this we ( * ) The η i are not integrals of the equations of motion (i.e. i ηɺ ). According to () the special choice of these quantities can only be valid for a certain time point. However this will suffice in what follows. Cf. P. A. M. Dirac Proc. Roy. Soc. (A) 8 (98) 5.

3 Möglich On the quantum theory of rotating electrons. observe that the coefficients of a matrix that admits a representation of the form S X S must be bilinear forms of the coefficients of S and S. However from the equation: α α η α (α ) () it follows immediately that the coefficients of α are linear forms of the rotation transformation coefficients a ik. From this it further follows that coefficients of the matrices S and S are so arranged that the a ik can be represented as bilinear forms in them. From now on one will then attempt to write out the a ik as bilinear forms of quadratic forms. One achieves this with the help of the Cayley-Klein parameters. The three-dimensional rotation group contains three parameters; e.g. the Euler angles ϑ ϕ ω. How one expresses the a ik in terms of them is well-known ( * ). One easily sees that this representation still does not give us the a ik as forms of second degree as desired. However if we set: ϑ i( ϕ+ ω )/ ϑ i( ϕ ω )/ α cos e β i sin e (5) ϑ i( ϕ ω) / ϑ i( ϕ+ ω )/ γ isin e γ cos e then this effortlessly yields the well-known representation of the Euler angles by the Cayley-Klein parameters: x x x i ( ) ( α β γ δ α β γ δ ) ( αγ βδ ) x + + i x α + β γ δ α + β + γ + δ i αγ + βδ x αβ γδ i( αβ + γδ ) αδ + βγ ( ) ( ) ( ) (6) If one observes () then one finds for the matrices α : i αγ βδ γ δ i α β αγ βδ αγ βδ γ δ α β αγ βδ i( ) i( ) i( ) i( ) ( ) i( ) ( ) i( ) ( * ) One finds the schema in e.g. Klein-Sommerfeld Über die Theorie des Kreisels pp. 9. Our x x x is identical with X Y Z resp. in it and our x x x are identical with x y z resp. The angle that is denoted by ω here is called ψ there.

4 Möglich On the quantum theory of rotating electrons. αγ + βδ γ + δ α + β αγ + βδ ( + ) ( + ) ( ) ( ) ( ) αγ βδ γ δ α + β αγ + βδ (7) i( αδ + βγ ) i γ δ i i( ) α β αδ + βγ. i( αδ + βγ ) i γ δ iα β i( αδ + βγ ) These matrices obviously decompose into two two-rowed matrices that differ from each other only by their signs. If we let ζ ζ ζ denote the two-rowed matrices: ζ i( αγ βδ ) i( γ δ ) i( α β ) i( αγ βδ ) ζ ζ i( αδ + βγ ) i γ δ iα β i( αδ + βγ ) αγ + βδ γ + δ ( ) ( ) α + β αγ + βδ and let ζ ζ ζ resp. denote the ones that result from the matrix equations (): ζ i i ζ ζ i i then we pose the problem of giving two-rowed transformation matrices P and P for which the equations: ζ P ζ P ζ P ζ P ζ P ζ P (8) are fulfilled. However this problem was already solved by Pauli ( * ). One easily verifies that the matrices: α γ P P δ γ (9) β δ β α satisfy equations (8). If one observes the relation that follows from equations (5): αδ βγ then one can easily show that one also actually has P P. ( * ) W. Pauli Zeit. f. Phys. (97) 6.

5 Möglich On the quantum theory of rotating electrons. 5 With the help of the matrices P and P we now define four-rowed matrices that take the η α [equation ()] to the α [equation (7)]. Namely one immediately shows that under the assumption of the validity of equations (8) the following equations are also true: R η R η R ηr η R ηr η R ηr η () if we understand R and R to mean the following matrices: R β δ α γ R β δ α γ γ δ α β. () γ δ α β If we now substitute () into equation () then because R and R commute with imc we can write R on the left of the entire equation. Since R is not zero the product can be zero only when the second factor vanishes and this means: h π mc η + η + π i x x h R ψ. (a) However this is equation () ( * ) with the following difference: The coordinates x x x are replaced with the primed coordinates by a spatial rotation of the former coordinate system and then the eigenfunctions are no longer the ψ but they have the form R ψ. If we have established that the functions ψ ψ ψ ψ are the eigenfunctions for some physical process in any coordinate system then the eigenfunctions in another coordinate system that arises from that one by rotation [transformation equations (6)] are: ψ β ψ + δ ψ ψ α ψ + γ ψ ψ β ψ δ ψ ψ α ψ + γ ψ. () From the matrix R [equation ()] one can then obviously obtain the matrix R by first going over to the matrix with complex-conugate elements and the switching the rows with the columns ( ** ). One calls a matrix that arises from another one in this way the adoint matrix to it; in general one denotes the adoint matrix by a. We then see that in our case the matrix R that is adoint to R and the matrix R that is reciprocal to R are identical to each other. One calls a transformation for which the matrix equation: R R () ( * ) The fact that the potentials are zero here is inessential since the result is true precisely when the Φ i are non-vanishing because the R and Φ i commute. ( ** ) One then has α * δ β * γ [cf. equation (5)].

6 Möglich On the quantum theory of rotating electrons. 6 is true unitary. These unitary transformations are the Hermitian analogues of realorthogonal transformations. Whereas e.g. for the real-orthogonal transformations the norm of a vector remains constant for unitary transformations this is the case for the sum of the absolute values of the components. In our case one easily sees that e.g.: J ψ kψ k () k is invariant under the transformation () a fact whose proof essentially stems from equation () when one observes that along with () the equations: ψ γ ψ + α ψ ψ δ ψ β ψ ψ γ ψ + α ψ ψ δ ψ β ψ are true. The invariance of the expression () under spatial rotations of the coordinate system allows one to interpret the invariant () as the density of the particle which is also very closely related as an analogy to the Born ( * ) and Pauli ( ** ) Anzsatz. The possibility of such an interpretation will then depend very crucially upon the behavior of the expression () under Lorentz transformations. Obviously the particle density must transform like the time component of a vector under a Lorentz transformation and furthermore the three spatial vector components must be found that will be extended to a four-current with the particle density that is then divergence-free. It might be the case that the expression () is no longer invariant under a Lorentz transformation. From that it would follow that the canonical transformation S η α S of the η α cannot be unitary if it corresponds to a Lorentz transformation. In the next section we will see that this condition is fulfilled and we will recognize that this condition is fulfilled only because of the fact that the Lorentz transformation is no longer real but represents an imaginary rotation. II. The transformations S T U and their three special Lorentz transformations. The Lorentz group contains six independent parameters corresponding to the displacements of the six two-dimensional coordinate planes in four dimensions. Among the six parameters one can choose three of them such that their variation leaves the time axis unchanged. This is the group of spatial rotations that we have already spoken of. A three-parameter subgroup then remains under which the time-axis is always transformed. We can compose this subgroup from three special Lorentz transformations namely the one in the x -direction the one in the x -direction and the one in the x -direction. Each of these transformations has the property that two of the three spatial axes remain unchanged. For the Lorentz transformation in the x -direction one has the coefficient matrix ( *** ): ( * ) M. Born Zeit. f. Phys. 8 (96) 8. ( ** ) W. Pauli loc. cit. pp. 66 equations (a) (b). ( *** ) Cf. M. von Laue Die Relatvitätstheorie vol. I sec. pp. 8 Braunschweig 99.

7 Möglich On the quantum theory of rotating electrons. 7 x x x x x x x x iβ β β x ix iβ β β v β c (5) If we introduce an imaginary angle Θ by way of the equations: then we get: iβ cos Θ sin Θ β β x x x x x cosθ sin Θ x x x sin Θ cosθ (5a) When one applies the transformation (5a) to equation () (assuming vanishing potentials Φ i ) it goes to the following equation: Here: and ( * ): h ψ ψ ψ ψ η + η + η + i mc ψ. (6) π i x x x x x i x η i η i a kη k. (7) k The matrices and then have the form: ( * ) The a k are now the coefficients of (5a). One then has: η η.

8 Möglich On the quantum theory of rotating electrons. 8 sin Θ i cos Θ sin Θ i cos Θ i cos Θ i sin Θ i cos Θ sin Θ cos Θ isin Θ cosθ isin Θ. isin Θ cos Θ i sin Θ cos Θ If one further observes that the coefficients of the primed matrices must be bilinear forms of the coefficients of the transformation matrices S and S then one comes to the representation of S and S in terms of the half-angle Θ /. One then easily finds that the matrices have the following form: One verifies that the equations: Θ Θ cos i sin Θ Θ cos isin S Θ Θ isin cos Θ Θ isin cos Θ Θ cos isin Θ Θ cos isin S Θ Θ i sin cos Θ Θ i sin cos S η S S η S. (8) are fulfilled but above all also the fact that the transformation S leaves η and η unchanged so one has: S η S η S η S η.

9 Möglich On the quantum theory of rotating electrons. 9 If the angle Θ were real then S would be identical with S as one sees immediately. However since the angle Θ is pure imaginary then S is a matrix with real coefficients. The transition to conugate-complex changes nothing and the switching of rows and columns yields: Θ Θ cos i sin Θ Θ cos isin S S Θ Θ isin cos Θ Θ isin cos and this matrix is completely different from S. We thus find it confirmed that the matrix transformation that corresponds to a Lorentz transformation is no longer unitary. The expression () is therefore not invariant under the special Lorentz transformation (5). However before we examine the behavior of the expression () closer we would first like to consider the other two special Lorentz transformations. If we subect equation () to the transformation: then (assuming Φ i ) it goes to: x x x x x x cos Θ sin Θ x x sin Θ cos Θ (9) h ψ ψ ψ ψ η + + η + i mc ψ. π i x x x x Only the matrices η and η will be changed; they go to: sin Θ cosθ sin Θ cos Θ cos Θ sin Θ cosθ sin Θ

10 Möglich On the quantum theory of rotating electrons. One immediately verifies that the matrices: cos Θ sin Θ cos Θ sin Θ. sin Θ cos Θ sin Θ cosθ satisfy the equations: Θ Θ cos sin Θ Θ cos sin T Θ Θ sin cos Θ Θ sin cos Θ Θ cos sin Θ Θ cos sin T Θ Θ sin cos Θ Θ sin cos () T η T T η T η T η T η T η T and are therefore the desired transformation matrices. Finally the Lorentz transformation: x x x x x x x cos Θ sin Θ x sin Θ cosθ () corresponds to the matrices:

11 Möglich On the quantum theory of rotating electrons. U U They satisfy the equations: i i Θ Θ Θ Θ i cos sin Θ Θ i Θ Θ sin i cos cos sin sin cos Θ Θ Θ Θ i cos sin. Θ Θ i Θ Θ sin i cos cos sin sin cos () η U η U η U η U U η U U η U in which: sin Θ i cos Θ sin Θ i cos Θ i cos Θ sin Θ i cos Θ sin Θ cos Θ isin Θ cosθ i sin Θ. i sin Θ cos Θ i sin Θ cos Θ None of the three matrices S T U are unitary. They would be so if the three quantities Θ Θ Θ were real angles. The quantities () are then unchanged under all Lorentz transformations. In the same way as we showed for the spatial rotations it follows that the eigenfunctions ψ transform under the transformations (5) (9) and () like Sψ Tψ Uψ resp. Naturally we can also combine the transformations (6) (5) (9) and () into a six-parameter transformation; the eigenfunctions then transform like:

12 Möglich On the quantum theory of rotating electrons. ψ RSTU ψ. Forming the products RSTU and (RSTU) U T S R is quite complicated and might therefore be omitted here. Remark added in proof: From a friendly remark of Wigner one can arrive at the fact that the matrices S T U assume the reduced form by a somewhat different choice of η i into which we can bring the matrix R. In order to do this as one easily confirms one needs only to replace the quantity η that we chose with: α. This is permissible since α also has the eigenvalues and and satisfies the equations: α η i + η i α in addition. III. The Lagrange function and the current vector. We replace x with x / i and Φ with i Φ in equation (). We further denote i η by η as we have already done. If we then multiply on the right by i η then this finally yields: h e i ηη + Φ ψ + mc η π i x c ψ. () For the sake of convenience we would like to write out the matrices η η η η and η explicitly. They are: η η i ε + i i + η η i ε + i + i + + η η i ε + i i + i i η η.

13 Möglich On the quantum theory of rotating electrons. We introduce an abbreviation: We shall understand ψ ε ψ to mean the scalar product of ψ * ( ψ ψ ψ ψ ) and the quantities (ε ψ); e.g.: ψ ε ψ i ψ ψ ψ ψ ψ ψ ψ ψ ( + ). ψ The quantities ψ ε then have a corresponding meaning. x k We now define the function: h ψ ψ h ψ ψ L m ψ ψ + ψ ε ψ ε π x x π i x x ie e + Φ ψ ψ + Φ αψ εαψ + mcψ ηψ. () c c We now demand that the integral: α L m dx dx dx dx must be an extremum when the ψ i and ψ i are considered to be independently-varied functions and the Euler-Lagrange equations become identical with equations () for the ψ i and the equation for the ψ i that one obtains from () when one replaces i with i ( ). The proof is carried out most simply by calculation. In the relativistic quantum mechanics that was developed by Gordon ( ) and Klein ( ) there is also a Lagrange potential whose variational equations are the equations of matter waves. A further and much more important feature of this theory was the fact that the derivatives of the Lagrange potentials represented the four components of the fourpotential of the electric current vector. We must see whether this is also true in the present case. Performing the differentiation of equation () yields ( ): ( ) The replacement of i with i is to be performed everywhere so it is also done in the matrices ε but not in x and Φ ; these quantities prove to be real. However because the i exists in the fourth term of the sum in () in such a way that one goes from x to x this i is not to be replaced with i. It is simplest for one to start with equation () replace all i with i and then go from x to x ; since η η this yields: h e h e i Φ ψ i ηη + Φ ψ π x c + πi x c η mc ψ *. ( ) W. Gordon Zeit. f. Phys. (96) 7. ( ) O. Klein ibidem (97). ( ) Naturally the expressions on the right are only instantaneous values of the current components. Cf. footnote ( ) on pp. 5 and the Remark added in proof on pp. 6.

14 Möglich On the quantum theory of rotating electrons. L e e Φ m ψ εψ [ ψ ψ + ψ ψ + ψ ψ + ψ ψ ] c c L e ie ψ ε ψ ψ ψ ψ ψ ψ ψ ψ ψ Φ m [ + ] c c Lm e e ψ εψ [ ψ ψ ψ ψ + ψ ψ ψ ψ ] Φ c c Lm ie ie ψ ψ [ ψ ψ + ψ ψ + ψ ψ + ψ ψ ]. Φ c c (5) If we now subect the wave equations () to the transformation (5a) then we now obtain the functions Sψ in place of the wave function ψ where S means the matrix (8). When written out the transformation looks like: Θ Θ Θ Θ ψ cos ψ + i sin ψ ψ cos ψ + isin ψ Θ Θ Θ Θ ψ i sin ψ + cos ψ ψ i sin ψ + cos ψ. (6) If we now form the expression (5) with the ψ i precisely as we did with the ψ i then we find after some calculations ( ): ψ εψ cos Θ ψ εψ + sin Θ i ψ ψ ψ ε ψ ψ ε ψ ψ ε ψ ψ ε ψ i ψ ψ sin Θ ψ εψ + cos Θ i ψ ψ (7) i.e. the four quantities (5) transform according to the Lorentz matrix (5a) and are therefore vector components. It still remains for us to show that that vector (5) is divergence-free. We differentiate each component of (5) with respect to its corresponding coordinate add e them after we have multiplied them by h / πi and finally add Φ ψ ε ψ + c ie ψ Φ ψ i mc ψ ηψ and subtract the same expressions. We then obtain: c h ψ ψ h ψ h ψ i ψ + ψ + ψ ε + ε ψ π i x x π i x π i x ( ) One observes the reality property that i sin Θ / is real.

15 Möglich On the quantum theory of rotating electrons. 5 + ie ie e Φ ψ ψ Φ ψ ψ + Φ ψ ε ψ c c c e mcψ η ψ ψ ε ψ ψ η ψ. Φ + mc c If we now combine the differential quotients ψ / x thus the ones that are multiplied by ψ * with the positive potential terms and the quantities mc ψ ηψ then we get when we set ε i: ψ h ε e + Φ ψ + η ψ mc. π i x c However this quantity vanishes because the right-hand factor agrees with (). We convert what remains slightly by observing that for the following equations are valid ( ): We then effortlessly get: ψ x ε ψ ψ ε ψ x ψ ε ψ ψ ε ψ. (8) h e h e ψ i Φ ψ + ε k Φkψ mcηψ π i x c k π i xk c. This expression likewise vanishes because the right-hand factor agrees precisely with the differential equations for the complex-conugate wave functions [footnote ( ) on pp. ]. The vanishing of the divergence of (5) is thus proved ( ). The invariance of the quantity L m can now be easily shown. Since the operator / x transforms like a vector what one finds in the curly bracket in () is essentially the scalar product of it and the vector (5). The second expression is the scalar product of the four-potential and the current vector while the invariance of ψ ηψ can be easily shown by calculation with the help of the transformations (8) () and (). Naturally this quantity is also a rotational invariant which comes from the fact that two invariants under rotations actually exist namely ψ + ψ and ψ + ψ. One then easily recognizes from equations () that ψ and ψ will transform amongst themselves as will ψ and ψ under a spatial rotation. Whereas the sum of these two rotational invariants is then ust rotationally-invariant the difference of the two is also Lorentz invariant. ( ) These equations are true for arbitrary Hermitian matrices. If the ε were no longer Hermitian then equations would be true in which one substitutes the transposes εɶ for the ε. The function Φ which is the solution of the transpose of equation () would then appear in place of ψ *. The function L m can also be generalized in this way. Cf. Dirac loc. cit. pp. 8. ( ) One finds another proof in Dirac loc. cit.

16 Möglich On the quantum theory of rotating electrons. 6 Remark added in proof: Since equations (8) are independent of any special choice of the ε we have proved the vanishing of the divergence of the current and the vector character of the quantities: ψ ε ψ ( ) and ψ ψ for an arbitrary time point. The invariance of L m for all time then follows from this. This result can be concluded from ust the Lorentz invariance of the Dirac equations. On the other hand the proof of the Lorentz invariance of the Dirac equation proceeds only under the assumption that p and the matrices R S T U commute with each other; i.e. they are constant in time. It first follows from this fact that the matrices R S T U that we obtained experience rotations and Lorentz transformations for all time. The fact that the derivatives of the Lagrange potential L m with respect to the components of the four-potential give the current components allows one to give a Lagrange function L under which not ust the Dirac equations but also Maxwell s equations can be preserved under variation of the ψ i the ψ i and the Φ i. One easily concludes that this function L has the form: L L m + 8π c (H E ). (9) One arrives at the energy-impulse tensor of matter waves in the same way that Schrödinger did. One can hardly confirm that this together with the tensor of the electromagnetic field yields a complete field theory but one can also not contradict it either. Certainly one will have to regard the ψ i -functions as q-number functions in the same way that Klein and Jordan ( * ) did. Naturally whether this alone will suffice to evade all of the difficulties with the many-body problem (e.g. the back-reaction of the proper field) cannot be decided without further information. The author hopes to return to these questions in the near future. ( * ) P. Jordan and O. Klein Zeit. f. Phys. 5 (97) 75.