The Dirac Sea, T and C Symmetry Breaking, and the Spinor Vacuum of the Universe^†

Round 1

Reviewer 1 Report

Summary
=========
This manuscript studies algebra and superalgebra of spinors, their Lorentz, charge, parity, and time reversal transformations, and their vacuum states. Although these subjects are not new, details presented in this work help clarify some of subtleties. Thus, the manuscript can be useful for both particle physics and condensed matter. However, significant corrections and modifications should be made before it can be accepted for publication. They are described in the following comments.

General
=======
- English style needs revision, especially in Abstract. Sec 1 & 2.
 
- The manuscript is very long. It can be somehow shortened by removing repetitions and trivial texts. For instance, in the paragraph after eq. (1) the author explains general formula for number of components of spinors and gives explicit values for several dimensions. This part can be shorten to just cases most useful for the rest of the discussion, namely n = 2 and 4.

- The word "hole" is employed in various places in the text as the remnant of the annihilation of a particle. However, The author should remind that this expression can be used only in the framework of a many particle system. For example, in an ionized atom, the absence of an electron does not generate a hole; When a photon decays to a pair particle and antiparticle, the latter is not called a hole. Moreover, a hole is not an antiparticle and is not necessarily a fermion (spinor). For instance, when a valence electron is displaced from a solid, its remnant is not an anti-electron (a positron) but a hole. In each case the correct expression must be used. 

Other comments
==============
line 86: It would be useful to remind that Lorentz transformation of complex functions and operators is not simply connected and C, T and P symmetries project them to one of other 3 disconnected spaces. Therefore, CPT conservation and Lorentz symmetry together present $O(3,1; C)$, as noticed e.g. in Ref [21].

line 111: The text correctly reminds that the vacuum must be CPT invariant, but may change under C, P, or T. This is similar to what happens in curved spacetime, where in general the vacuum defined as the state annihilated by annihilation operator does not remain a vacuum under Lorentz transformation. This observation shows the ambiguity of vacuum definition, see e.g. Birrell and Davies, "Quantum fields in curved spacetime", (1983). These properties of vacuum state should be discussed in the manuscript.

line 135: Whatever the reason for calling spinors 'invariants" in old texts, the name is confusing and useless for the discussion in this work. Please remove it.

line 203: In the sentence beginning with "It turns out ....": This sentence does not make sense. Momentum is not either a spinor or a gauge field. The sentence must be corrected or removed.

lines 205-209, paragraph beginning with "In the case of ....": This paragraph needs the  definition of operators $b$ and $\bar{b}$. It is better to transfer it to Sec. 3 where the algebra is defined and add missing components.

line 336, Sec 4: The author should briefly explain what is the goal of this section and why for achieving this goal momentum should be discretized.

lines 348-358: This paragraph somehow answers the previous comment and as suggested above can be moved to the beginning of Sec. 4. However, a couple of issues remain: Discretization is not needed for defining a Hilbert space. It can be continuous and number of states uncountable. Although a countable Hilbert space is mathematical better defined and easier to manipulate, in a physical context an analytical continuation is more meaningful and convenient. Moreover, the set of fractional numbers is a dense subspace of real number. Therefore, an analytical extension to continuity usually exists. Emphasizing on discretization may be confusing and give the impression that it is part of the physics of fermions. These technicalities should be explained. Indeed, in the following sections continuous variables are used in the formulation. Therefore, lengthy discussion about discretization is not necessary. 

lines 396-406: The claim of infinite number of vacuums is not correct. In the definition of $\Psi_v(P_i)$ according to eqs. (36) and (37) changing the order of pair $b_i \bar{b_i}$ simply adds a $-1 = e^i\pi$ phase factor to the vacuum state. This factor is irrelevant because in quantum mechanics states are rays. For the same reason the order of $\Psi_v(pi)$ factors in eq. (32) is not important. It adds at most an irrelevant sign factor to the definition of vacuum state. On the other hand,  the role of boost angles \phi_k is not sufficiently discussed. They are in general momentum dependent, change the order of p_i factors in eq (32), and multiply the state by an over all phase. This phase would also be irrelevant, but both phases may become relevant if fermions/spinors have other charges, e.g. flavor, and one needs to write a superposition of vacua with different flavors. Proper discussion of these points is very important for the manuscript because in the abstract the author has considered them as new results of this work. Accordingly, the abstract must be modified.

lines 407-413: This paragraph is not clear. Please remove it or clarify what are "additional spinor columns", etc.

lines 418-420: The paragraph is a repetition. Please remove it.

Eq.(49): The left hand side is not correct. Boost operator must be applied to both $b$ and $\bar{b}$, as in eq.(52).

Eq.(54): On the r.h.s. order of operators must be inverted.

lines 468-474: As explained before, changing order of $b$ and $\bar{b}$ operators does not change the vacuum - up to an irrelevant phase factor. This paragraph reminds that flipping these operators is equivalent to reversing the definition of annihilation and creation operators. They can be also interpreted as operators for creation and annihilation of antiparticles. But in both interpretations the vacuum does not include any particle or antiparticle. Thus, the interpretation of eq.(61) must be corrected.

line 807: Section number must be corrected. Moreover, the content of this section should be summarizing the results not repeating previous discussions. It should be significantly shortened and formulas which have been presented elsewhere in the manuscript must be removed. As mentioned earlier, there are too many repetitions in the manuscript. They must be removed to make the text more concise.

Author Response

1.> English style needs revision, especially in Abstract. Sec 1 & 2.

 - Revision done.

2.> The manuscript is very long. It can be somehow shortened by removing repetitions and trivial texts. For instance, in the paragraph after eq. (1) the author explains general formula for number of components of spinors and gives explicit values for several dimensions. This part can be shorten to just cases most useful for the rest of the discussion, namely n = 2 and 4.

- The manuscript has been substantially reduced, from 31 pages to 24 pages.

3.> The word "hole" is employed in various places in the text as the remnant of the annihilation of a particle. However, The author should remind that this expression can be used only in the framework of a many particle system. For example, in an ionized atom, the absence of an electron does not generate a hole; When a photon decays to a pair particle and antiparticle, the latter is not called a hole. Moreover, a hole is not an antiparticle and is not necessarily a fermion (spinor). For instance, when a valence electron is displaced from a solid, its remnant is not an anti-electron (a positron) but a hole. In each case the correct expression must be used. 

- This remark is, on the whole, quite correct. However, in this article, the term "hole" is always used in the context of the concept of the Dirac Sea. Within the framework of this concept, the concept of antiparticles came into being. In the concept of the Sea of Dirac, the "hole" in this Sea is always an antiparticle. Due to the fact that there were complaints about the large size of the article, I think it is better not to add the proposed explanations to the article.

Other comments
==============
4.>line 86: It would be useful to remind that Lorentz transformation of complex functions and operators is not simply connected and C, T and P symmetries project them to one of other 3 disconnected spaces. Therefore, CPT conservation and Lorentz symmetry together present $O(3,1; C)$, as noticed e.g. in Ref [21].

- Correction done (text at the bottom of page 2 and beginning of page 3).

5.>line 111: The text correctly reminds that the vacuum must be CPT invariant, but may change under C, P, or T. This is similar to what happens in curved spacetime, where in general the vacuum defined as the state annihilated by annihilation operator does not remain a vacuum under Lorentz transformation. This observation shows the ambiguity of vacuum definition, see e.g. Birrell and Davies, "Quantum fields in curved spacetime", (1983). These properties of vacuum state should be discussed in the manuscript.

- This is a very interesting issue. A short discussion has been added at the end of the "Discussion" section. However, a detailed discussion of this issue goes far beyond the topic of this article and requires additional research.

6.>line 135: Whatever the reason for calling spinors 'invariants" in old texts, the name is confusing and useless for the discussion in this work. Please remove it.

- Corresponding text has been removed.

7.>line 203: In the sentence beginning with "It turns out ....": This sentence does not make sense. Momentum is not either a spinor or a gauge field. The sentence must be corrected or removed.

- Corresponding text has been removed.

8.>lines 205-209, paragraph beginning with "In the case of ....": This paragraph needs the  definition of operators $b$ and $\bar{b}$. It is better to transfer it to Sec. 3 where the algebra is defined and add missing components.

- Correction done, corresponding text after formula (20) and formula (21) were added to Sec. 2 (before corrections it was Sec. 3).

9.>line 336, Sec 4: The author should briefly explain what is the goal of this section and why for achieving this goal momentum should be discretized.

>lines 348-358: This paragraph somehow answers the previous comment and as suggested above can be moved to the beginning of Sec. 4. However, a couple of issues remain: Discretization is not needed for defining a Hilbert space. It can be continuous and number of states uncountable. Although a countable Hilbert space is mathematical better defined and easier to manipulate, in a physical context an analytical continuation is more meaningful and convenient. Moreover, the set of fractional numbers is a dense subspace of real number. Therefore, an analytical extension to continuity usually exists. Emphasizing on discretization may be confusing and give the impression that it is part of the physics of fermions. These technicalities should be explained. Indeed, in the following sections continuous variables are used in the formulation. Therefore, lengthy discussion about discretization is not necessary. 

- Correction done. However, it was not possible to completely remove the words about discretization, since in order to construct a spinor vacuum, it is necessary to consider the product of pairs of annihilation operators and creation operators for all possible values of momentum. And the elements of such a product, by definition, form a countable set.

10.>lines 396-406: The claim of infinite number of vacuums is not correct. In the definition of $\Psi_v(P_i)$ according to eqs. (36) and (37) changing the order of pair $b_i \bar{b_i}$ simply adds a $-1 = e^i\pi$ phase factor to the vacuum state. This factor is irrelevant because in quantum mechanics states are rays.

- Apparently, in the text of the manuscript it was not clearly stated that it is not the pairs $b_i \bar{b_i}$  that change places, but $b_i$ and $\bar{b_i}$  within the corresponding pair $b_i \bar{b_i}$. In the definition of $\Psi_v(P_i)$ according to eq. (36) (eq. (25) in the corrected version of the article) changing the order of $b_i$ and  $\bar{b_i}$ does not add a $-1 = e^i\pi$ phase factor to the vacuum state. Corresponding corrections, making the text more understandable, were made in the corrected text after formula (35) in the corrected version of the article.

Additional comment: in the QFT, because of the nonzero value on the right-hand side of eq. (36) (eq. (25) in the corrected version of the article), infinite additional terms appear. Usually they get rid of them by normalizing the order of the creation and annihilation operators. In this approach, the creation and annihilation operators are considered anticommuting. In the theory of superalgebraic spinors, infinite constants do not arise, since the gamma operators and other operators constructed from them, including the Hamiltonian, are commutators. In this case, we construct the vacuum state vector from the products of the annihilation operators by the creation operators. And we don't use switches. Therefore, we cannot assume that the annihilation operators anticommute with the creation operators.

11.>For the same reason the order of $\Psi_v(pi)$ factors in eq. (32) is not important. It adds at most an irrelevant sign factor to the definition of vacuum state.

- This statement is true, these factors commute.

12.>On the other hand,  the role of boost angles \phi_k is not sufficiently discussed. They are in general momentum dependent, change the order of p_i factors in eq (32), and multiply the state by an over all phase. This phase would also be irrelevant, but both phases may become relevant if fermions/spinors have other charges, e.g. flavor, and one needs to write a superposition of vacua with different flavors.

- Factors corresponding to different momentums commute. Therefore, as a result of the Lorentz transformation, there is no change in the phase of $\Psi_v$. In the corrected version of the article, corresponding correction was made to the text after formula (42).

If there are several kinds of spinors, the resulting spinor vacuum is the product of the spinor vacuums for each of the kinds of spinors. Lorentz transformations do not mix vacuums of different kinds of spinors. However, operators mixing different kinds of spinors are possible. The problem of mixing different kinds of spinors and the corresponding vacuum transformations are beyond the scope of this article and require additional research.

13.>Proper discussion of these points is very important for the manuscript because in the abstract the author has considered them as new results of this work. Accordingly, the abstract must be modified.

- As already mentioned, there were no errors in the article, but an incomprehensibly stated formulation of the reason for the emergence of alternative vacuums. The corresponding corrections were done in the revised text after formula (35).

14.>lines 407-413: This paragraph is not clear. Please remove it or clarify what are "additional spinor columns", etc.

- Corresponding text has been removed.

15.>lines 418-420: The paragraph is a repetition. Please remove it.

- Corresponding text has been removed.

16.>Eq.(49): The left hand side is not correct. Boost operator must be applied to both $b$ and $\bar{b}$, as in eq.(52).

- The left-hand side is correct because the Lorentz transformation operator is R-operator. The corresponding explanations are made in the corrected version of the article in the text before formula (42) and in this formula.

17.>Eq.(54): On the r.h.s. order of operators must be inverted.

- In the revised version of the article, this is formula (45). According to (9)-(11), $\bar{\Psi’}=(\gamma^0 \gamma^0\gamma^5 \Psi)^+ = - \gamma^0 (\gamma^0\gamma^5 \Psi)^+ =  \gamma^0 (\gamma^5 \gamma^0\Psi)^+ =

= - \gamma^0 \gamma^5 (\gamma^0\Psi)^+ $.

In the original version of the article, the minus sign was forgotten on the right-hand side; in the corrected version, the typo was corrected.

18.>lines 468-474: As explained before, changing order of $b$ and $\bar{b}$ operators does not change the vacuum - up to an irrelevant phase factor. This paragraph reminds that flipping these operators is equivalent to reversing the definition of annihilation and creation operators. They can be also interpreted as operators for creation and annihilation of antiparticles. But in both interpretations the vacuum does not include any particle or antiparticle. Thus, the interpretation of eq.(61) must be corrected.

- Explanation was given in response to Remark 10. The statements in this paragraph are correct and do not need to be changed.

19.>line 807: Section number must be corrected. Moreover, the content of this section should be summarizing the results not repeating previous discussions. It should be significantly shortened and formulas which have been presented elsewhere in the manuscript must be removed. As mentioned earlier, there are too many repetitions in the manuscript. They must be removed to make the text more concise.

- All corrections recommended by the reviewer have been made.

Reviewer 2 Report

The author developed a superalgebraic representation of spinors using of Grassmann densities in the momentum space.  It have proved that CPT is the real structure operator in the theory of Krein spaces. The notion of an alternative vacuum is invastigated, and it is shown that C and T operators transform a normal vacuum into an alternative one, which leads to the breaking of these symmetries. The connection between the formulas for the concept of "holes" in the Dirac Sea and the algebraic theory of spinors is established.

Author Response

There are no comments from the reviewer to be answered.

Reviewer 3 Report

The paper turned out to be somewhat outside my area of expertise. While the technical part seems sound and intersting, the author and editor should not rely too heavily on my expertise here. 

Instead, I would just recommend improving the structure and overall presentation. The introduction is full of "historical" references but it doesn't become quite clear to me what problem, specifically, the paper wants to address and how its results differ from established ways to implement C, P, T symmetries. Also: to what extent is the approach based on the theory of superalgebraic spinors just a different formulation of "standard QFT" versus a physically distinct framework?

Finally, the paper seems somewhat long. The author might consider shorten it and/or delegating some computations to an Appendix, if appropriate. 

Author Response

>I would just recommend improving the structure and overall presentation.

- Thank you for your comments. I tried to correct the text of the article in accordance with these comments.

>The introduction is full of "historical" references but it doesn't become quite clear to me what problem, specifically, the paper wants to address and how its results differ from established ways to implement C, P, T symmetries.

- The "historical" links have been significantly shortened and hopefully the text has become clearer. The main novelty of the article is indicated in its title and abstract.

>Also: to what extent is the approach based on the theory of superalgebraic spinors just a different formulation of "standard QFT" versus a physically distinct framework?

- The approach based on the theory of superalgebraic spinors is a development of algebraic quantum field theory. It differs from the usual relativistic QFT only in the use of a more developed mathematical apparatus. In fact, this approach develops the ideas and formulas proposed by Julian Schwinger in the article Schwinger, J. The Theory of Quantized Fields. I. Phys. Rev. 1951, 82, 914–927.

Modern mathematical results from the theory of algebraic spinors and Clifford algebras are added to Schwinger's approach.

>Finally, the paper seems somewhat long. The author might consider shorten it and/or delegating some computations to an Appendix, if appropriate. 

-  The paper has been substantially reduced, from 31 pages to 24 pages.

Round 2

Reviewer 1 Report

Modifications are adequate.