Quaternion Electromagnetism and the Relation with Two-Spinor Formalism

Round 1

Reviewer 1 Report

This ms represents an interesting and potentially valuable extension of quaternion calculus by letting the field of this vector space change from R  to C. This entails a compact way of writing Maxwell's equations and studying their and the covariance of their fields and composites under continuous and discrete spacetime transformations. Also, the authors consider higher-dimensional generalisations and the connection between spacetime and spin variables. To grasp the notations needs a bit of training, but finally they appear to make sense, and they are efficient. I like the statements about parity and Lorentz transformations which become impressively compact in the complex quaternion formulation. Is it always true that in EM theory the real (imaginary) part is parity(P)  even (odd)?  What about T or CP? The contemplations in Sec. VIII are quite speculative, but since this is stated explicitely, there is no problem here. 

Some minor points of the typo type:

1) In Eq. (7) the factorial signs (!) are missing in the denominators!

2) a electric -> an electric

3) Below (37): The inverse and ... -> The inverse of \zeta and its complex conjugate ...

4) last sentence on p. 9: Terminate that sentence after `boosts´. The remainder cannot be understood and is almost tautological. 

5) In general, try to throw a few more articles (a, the) into your sentences. 

After these minor points are corrected I recommend the publications of this ms in Universe. I don't need to see the ms again.

Author Response

Thank you for your kind review.

We have modified as follows.

1) In Eq. (7) the factorial signs (!) are missing in the denominators!

—> We found that your requests are correct, we modified the details as you mentioned.

2) a electric -> an electric

---> We found that your requests are correct, we modified the details as you mentioned.

3) Below (37): The inverse and ... -> The inverse of \zeta and its complex conjugate ...

---> We found that your requests are correct, we modified the details as you mentioned.

4) last sentence on p. 9: Terminate that sentence after `boosts´. The remainder cannot be understood and is almost tautological.

---> We found that your requests are correct, we modified the details as you mentioned.

5) In general, try to throw a few more articles (a, the) into your sentences.

---> We corrected as you advised.

Reviewer 2 Report

There are a number of comments:

1) Page 5, 3th line after Eq.(11):  We read "J is equal to rho * u" then according to the first formula of Eq.(11), the fourth formula should contain the factor gamma, namely, it should be written J = gamma * rho + i * gamma * {\vec J}. Check it, please.

2) Page 5, explanation of terms in Eq.(11): Among the list of explanations of the formulas given in Eq.(11) there is no explanation of the term of the fifth formula.

3) Page 5, Eq.(11): One can see that the equations in lines (6) and (7) are equivalent. It is not possible to understand, why should lines (6) and (9) be equal? This equality is written in the third line after Eq.(28) in page 7.

4) Page 6, the first line: "but the others have not been known so far". This statement is not true. Relations (2), (5) are known, see, for example, Wikipedia: Lorenz gauge condition - https://en.wikipedia.org/wiki/Lorenz_gauge_condition ; Inhomogeneous electromagnetic wave equation - https://en.wikipedia.org/wiki/Inhomogeneous_electromagnetic_wave_equation (compare Eqs.(21) with those given in this Wikipedia page)

5) Page 8, Eq.(31): in the first matrix, S(Lambda_rot), the forth matrix element should have sign minus at the imaginary unit in exponent. Check it, please.

6) Page 12, Eq.(52): what does the symbol epsilon^{\mu\nu\rho\sigma} in Eq.(52)? Give the definition, please.

7) Page 13, line below Eq.(58):  the complexified electromagnetic tensor F^{tilda mu,nu}  +  i * G^{tilda mu,nu}  is known in literature, see Girard, P. R., "Quaternions, Clifford Algebras and Relativistic Physics", (Birkh\"auser, Basel, 2000), subsec. 3.6 Maxwell’s equations, page 50; Sbitnev, V. I., "Hydrodynamics of superfluid quantum space: particle of spin-1/2 in a magnetic field", Quantum Stud.: Math. Found., pp. 297-314, v.5(2), 2018; Sbitnev, V. I., "Quaternion Algebra on 4D Superfluid Quantum Space-Time: Gravitomagnetism", Foundations of Physics, pp.107-143, v.49(2), 2019. References to these works are desirable. They will confirm your awareness of such directions of investigations  and show that you are not alone in these conclusions.     

8) Page 14, fourth formula in Eq.(61): As I understand this tensor is very important. What is name of this tensor?  Give, please, explanation of all components of this tensor. I mean the first diagonal element u, as well as the nondiagonal element p with referring to Eq.(8).

9) Page 14, Eq.(64): As I understand from Eqs.(59),(60),(61), the brackets should contain the expression D^*{Gothic F}^* = J^*. Check it please.

10) Page 15, the first line below Eq.(74): what does the symbol epsilon^{CB}? Give the definition, please.

11) Page 15, the second line below Eq.(75): `Infeld-van der Waerden symbol'. It is desirable to specify the source of this expression: L. Infeld, B.L. Van der Waerden, "Die wellengleichung des elektrons in der allgemeinen relativit\"atstheorie" Sitzber. Preuss. Akad. Wiss., Physik-Math.K1. v.9, pp.380-401 (1933)  
                                                                                                          
12) Page 19,  Eqs.(103) and (107): according to the multiplication rule of quaternions given in Eq.(1), the rightmost part of Eq.(103) is (-i,-1.-k, j), the rightmost part of Eq.(107) is (-i, k,-1,-j). Check it please.                                                                                                   

Comments for author File: Comments.pdf

Author Response

Thank you for your kind review.

We changed as follows. (We added a Eq (25) so the Equation numbers are slightly changed)

For your convenience, we used modified parts as red color in the manuscript.

1) Page 5, 3th line after Eq.(11): We read "J is equal to rho * u" then according to the first formula of Eq.(11), the fourth formula should contain the factor gamma, namely, it should be written J = gamma * rho + i * gamma * {\vec J}. Check it, please.

>>>>>>>>>>>[ We modified the parts in page 5.]

2) Page 5, explanation of terms in Eq.(11): Among the list of explanations of the formulas given in Eq.(11) there is no explanation of the term of the fifth formula.

>>>>>>>>>>>[ We checked the parts that you have mentioned, and we modified the parts in page 5.]

3) Page 5, Eq.(11): One can see that the equations in lines (6) and (7) are equivalent. It is not possible to understand, why should lines (6) and (9) be equal? This equality is written in the third line after Eq.(28) in page 7.

>>>>>>>>>>>>>[ The relation 7) is focused on the nontrivial associative rule of F d \bar{F}. We added parenthesis in Eq.(13) and the third line on page 7.

Following your advice we added a comment of the possibility that 6) and 9) are equal on page 7. ]

4) Page 6, the first line: "but the others have not been known so far". This statement is not true. Relations (2), (5) are known, see, for example, Wikipedia: Lorenz gauge condition - https://en.wikipedia.org/wiki/Lorenz_gauge_condition ; Inhomogeneous electromagnetic wave equation - https://en.wikipedia.org/wiki/Inhomogeneous_electromagnetic_wave_equation (compare Eqs.(21) with those given in this Wikipedia page)

>>>>>>>>>>>>>[ Those relations in the quaternion relations (2),(5) are already known, however, what we have originally meant was that the representations of them in 'quaternion form' were not known. The relation 6) is a kind of an application of the quaternion relation 4) since it  can be derived trivially from taking a differential operator \d{d} on both side of the relation 4).

We revised the sentence below Eq (13), and added a comment under Eq. (21). ]

5) Page 8, Eq.(31): in the first matrix, S(Lambda_rot), the forth matrix element should have sign minus at the imaginary unit in exponent. Check it, please.

>>>>>>>>>>>>>[ We have checked and found that there is no problem.  Please see (4.58) in http://www.damtp.cam.ac.uk/user/tong/qft/four.pdf ]

6) Page 12, Eq.(52): what does the symbol epsilon^{\mu\nu\rho\sigma} in Eq.(52)? Give the definition, please.

>>>>>>>>>>>>>[ As you requested, we added an explanation below Eq.(53) ]

7) Page 13, line below Eq.(58): the complexified electromagnetic tensor F^{tilda mu,nu} + i * G^{tilda mu,nu} is known in literature, see Girard, P. R., "Quaternions, Clifford Algebras and Relativistic Physics", (Birkh\"auser, Basel, 2000), subsec. 3.6 Maxwell’s equations, page 50; Sbitnev, V. I., "Hydrodynamics of superfluid quantum space: particle of spin-1/2 in a magnetic field", Quantum Stud.: Math. Found., pp. 297-314, v.5(2), 2018; Sbitnev, V. I., "Quaternion Algebra on 4D Superfluid Quantum Space-Time: Gravitomagnetism", Foundations of Physics, pp.107-143, v.49(2), 2019. References to these works are desirable. They will confirm your awareness of such directions of investigations and show that you are not alone in these conclusions.

>>>>>>>>>>>>>[ Following  your advice, we read all  those references and we added them as citations below Eq.(59) ]

8) Page 14, fourth formula in Eq.(61): As I understand this tensor is very important. What is name of this tensor? Give, please, explanation of all components of this tensor. I mean the first diagonal element u, as well as the nondiagonal element p with referring to Eq.(8).

>>>>>>>>>>>>>[ As  you requested ,we added Eq. (28), reduced the explanation below Eq. (62) and changed the explanation of T as below Eq. (70).

9) Page 14, Eq.(64): As I understand from Eqs.(59),(60),(61), the brackets should contain the expression D^*{Gothic F}^* = J^*. Check it please.

>>>>>>>>>>>>>[ We have checked and found that there is no problem in (64) – now renamed as (65).  {Gothic F}^* is not just complex conjugate of (Gothic F), as shown in (60,61). ]

10) Page 15, the first line below Eq.(74): what does the symbol epsilon^{CB}? Give the definition, please.

>>>>>>>>>>>>>[ We added the explanations of 'epsilon' below Eq.(75)]

11) Page 15, the second line below Eq.(75): `Infeld-van der Waerden symbol'. It is desirable to specify the source of this expression: L. Infeld, B.L. Van der Waerden, "Die wellengleichung des elektrons in der allgemeinen relativit\"atstheorie" Sitzber. Preuss. Akad. Wiss., Physik-Math.K1. v.9, pp.380-401 (1933)

>>>>>>>>>>>>>[ We added the citation below Eq.(76)]

12) Page 19, Eqs.(103) and (107): according to the multiplication rule of quaternions given in Eq.(1), the rightmost part of Eq.(103) is (-i,-1.-k, j), the rightmost part of Eq.(107) is (-i, k,-1,-j). Check it please.

>>>>>>>>>>>>>[ We  found that you are correct and changed  Eq.(104) and Eq.(108)]

Further, we changed Equation reference on Eq (71), and we add citation at the beginning of Section VII.　A.