Correction: Buchbinder, I.L.; Reshetnyak, A.A. Covariant Cubic Interacting Vertices for Massless and Massive Integer Higher Spin Fields. Symmetry 2023, 15, 2124

The authors wish to make the following corrections in their paper [1]:

1. Additional Affiliations

In the published publication, there was an error regarding the affiliation(s) for A. A. Reshetnyak. The corrected affiliations should be updated to “2,3,5,*”. In addition to affiliation 2, the updated affiliation should include: Center of Theoretical Physics, Tomsk State Pedagogical University, 634061 Tomsk, Russia. Add an email address for A. A. Reshetnyak which is [email protected], and keep the both email addresses in the paper. In addition to affiliation 5, the updated affiliation should include: Department of Experimental Physics, National Research Tomsk Polytechnic University, 634050 Tomsk, Russia. The authors state that the scientific conclusions are unaffected. This correction was approved by the Academic Editor. The original publication has also been updated.

2. The last sentence in the Introduction should be extended by the additional notation used in [1]:

⋯, ${\theta }_{k,l}$$=1\left(0\right)$, when $k>l(k\le l)$, ${\left(s\right)}_3$ is the Heaviside $\theta$-symbol.

3. The sentence after Equation (20) should be modified to the following:

We prove in Appendix A that after imposing the appropriate gauge conditions and eliminating the auxiliary fields with the help of the equations of motion, the theory under consideration is reduced to an ungauged form equivalent to the Singh–Hagen action [46] in terms of a totally symmetric traceful tensor field ${\varphi }^{\mu \left(s\right)}$ and auxiliary traceful ${\varphi }_1^{\mu (s-3)}$.

4. After Equation (138), the following sentence and formula expression should be inserted:

Then, we use the notation for the l-th trace degree ${{\varphi }_{0,0}^{\left(3\right)\mu \left(s-2l\right){\rho }_1\dots {\rho }_l}}_{{\rho }_1\dots {\rho }_l}\equiv {\varphi }_{0,0}^{\left(3\right)\mu \left(s-2l\right)}$ and the symmetrizer (B.4), (B.5) $S_{\mu \left(k\right)}^{\nu \left(k\right)}\equiv S_{{\mu }_1\dots \dots {\mu }_k}^{{\nu }_1\dots {\nu }_k}$, with the property ${\prod }_{j=1}^k{a}^{+{\nu }_j}={\displaystyle \frac{1}{k!}}{S}_{\mu \left(k\right)}^{\nu \left(k\right)}{\prod }_{i=1}^k{a}^{+{\mu }_i}$, to present the vector (A31) $|{\varphi }_{2k}{\rangle }_{s-2k}$ in the tensor form with definite rationals ${\alpha }_{k|l}^0$ for $k\ge 1$:

$${\varphi }_{0,2k}^{\left(3\right)\nu (s-2k)}={(-1)}^k\sum _{l=1}^{[s-1/2]-k}{\displaystyle \frac{{\alpha }_{k|l}^0}{(s-2(k+l\left)\right)!}}{S}_{\mu (s-2k)}^{\nu (s-2k)}\prod _{t=1}^l{\eta }^{{\mu }_{s-2(k+l-t)-1}{\mu }_{s-2\left(\right(k+l-t)}}{\varphi }_{0,0}^{\left(3\right)\mu (s-2k-2l)}.$$

5. In the sentence after Equation (138), the expression “⋯ and auxiliary ${\varphi }_{0,3}^{\left(3\right)\nu \left(s-3\right)}$ fields ⋯” should be changed to the following:

“⋯ and auxiliary ${\varphi }_{0,1}^{\left(3\right)\nu \left(s-3\right)}\equiv {\tilde{\varphi }}_{0,1}^{\left(3\right)\nu \left(s-3\right)}$ fields ⋯”

6. Equation (140) and the comment that follows should be changed to

$$\begin{array}{ccc}& & S_{{1|\left(s\right)}_3}^{{\left(m\right)}_3}\left[{\varphi }^{\left(i\right)},{\varphi }^{\left(3\right)}\right]=-2g\int d^{d}x\left[\sum _{j\ge 0}^{[s/2]}{t}_j\sum _{k\ge 0}^j{\displaystyle \frac{{(-1)}^{k}j!(s-2j)!}{2^{2(j-k)}k!(j-k)!}}\right\{\sum _{u=0}^{s-2j}{\displaystyle \frac{{(-1)}^u}{u!(s-2j-u)!}}\hfill \\ & & \times \left[{\partial }_{{\nu }_0}\dots {\partial }_{{\nu }_u}{\varphi }^{\left(1\right)}\right]\left[{\partial }_{{\nu }_{u+1}}\dots {\partial }_{{\nu }_{s-2j}}{\varphi }^{\left(2\right)}\right]\left\}\prod _{r=1}^{j-k}{\eta }_{{\nu }_{s-2j+2r-1}{\nu }_{s-2j+2r}}\right({\varphi }_{0,0}^{\left(3\right)\mu \left(s\right)}{\delta }_{k,0}\hfill \\ & & +\sum _{l=1}^{[s-1/2]-k}{\displaystyle \frac{{\alpha }_{k|l}^0{S}_{\mu (s-2k)}^{\nu (s-2k)}}{(s-2(k+l\left)\right)!}}\prod _{t=1}^l{\eta }^{{\mu }_{s-2(k+l-t)-1}{\mu }_{s-2(k+l-t)}}{\varphi }_{0,0}^{\left(3\right)\mu (s-2k-2l)})\hfill \\ & & +\sum _{j\ge 0}^{[s/2]-1}{t}_j\sum _{k\ge 0}^j{\displaystyle \frac{{(-1)}^k(j+1)!(s-2(j+1))!}{2^{2(j-k)}k!(j-k)!}}\{\sum _{u=0}^{s-2(j+1)}{\displaystyle \frac{{(-1)}^u}{u!(s-2(j+1)-u)!}}\left[{\partial }_{{\nu }_0}\dots {\partial }_{{\nu }_u}{\varphi }^{\left(1\right)}\right]\hfill \\ & & \times \left[{\partial }_{{\nu }_{u+1}}\dots {\partial }_{{\nu }_{s-2(j+1)}}{\varphi }^{\left(2\right)}\right]\left\}\prod _{r=1}^{j-k}{\eta }_{{\nu }_{s-2(j+1)+2r-1}{\nu }_{s-2(j+1)+2r}}\right(-{\varphi }_{0,0}^{\left(3\right)\mu \left(s-2\right)}{\delta }_{k,0}\hfill \\ & & +\sum _{l=1}^{[s-3/2]-k}{\displaystyle \frac{{\alpha }_{k+1|l}^0{S}_{\mu (s-2(k+1\left)\right)}^{\nu (s-2(k+1\left)\right)}}{(s-2(k+l+1\left)\right)!}}\prod _{t=1}^l{\eta }^{{\mu }_{s-2(k+1+l-t)-1}{\mu }_{s-2(k+1+l-t)}}{\varphi }_{0,0}^{\left(3\right)\mu (s-2k-2-2l)}\hfill \\ & & -{\eta }_{{\nu }_{s-2k}{\nu }_{s-2k-1}}\sum _{l=1}^{[s-1/2]-k}{\displaystyle \frac{{\alpha }_{k|l}^0{S}_{\mu (s-2k)}^{\nu (s-2k)}}{(s-2(k+l\left)\right)!}}\prod _{t=1}^l{\eta }^{{\mu }_{s-2(k+l-t)-1}{\mu }_{s-2(k+l-t)}}{\varphi }_{0,0}^{\left(3\right)\mu (s-2k-2l)}\left)\right]\hfill \end{array}$$

(140)

(for ${\mathcal{S}}_{C|s}^m\left[\varphi ,{\varphi }_1\right]$ defined in (A32), (A33), with definite rationals ${\alpha }_{k|l}^0$ from (A31), without involving the field ${\varphi }_{0,1}^{\left(3\right)\nu \left(s-3\right)}$ into interaction) determines ungauge theory with accuracy up to the first order in g.

7. The sentence before Equations (141) and (142) should be corrected as follows:

For spin $s=4$, the interacting action (140) for unconstrained massive field ${\varphi }_{0,0}^{\left(3\right)\nu \left(4\right)}$ (for ${\varphi }_{0,2}^{\left(3\right)\nu \left(2\right)}=-{\displaystyle \frac{2}{d}}{\eta }^{\nu \left(2\right)}{{\varphi }_{0,0}^{\left(3\right)\mu \nu }}_{\mu \nu }$) takes the form

$$\begin{array}{ccc}& & S_{1\left|\right(0,0,4)}^{(0,0,m)}\left[{\varphi }^{\left(i\right)},{\varphi }^{\left(3\right)}\right]=-2g\int d^{d}x\{t_0{\varphi }_{0,0}^{\left(3\right)\nu \left(4\right)}\left({\varphi }^{\left(1\right)}{\partial }_{{\nu }_1}\dots {\partial }_{{\nu }_4}-4\left[{\partial }_{{\nu }_1}{\varphi }^{\left(1\right)}\right]{\partial }_{{\nu }_2}\dots {\partial }_{{\nu }_4}\right.\hfill \\ & & +6\left[{\partial }_{{\nu }_1}{\partial }_{{\nu }_2}{\varphi }^{\left(1\right)}\right]{\partial }_{{\nu }_3}{\partial }_{{\nu }_4}-4\left[{\partial }_{{\nu }_1}{\partial }_{{\nu }_2}{\partial }_{{\nu }_3}{\varphi }^{\left(1\right)}\right]{\partial }_{{\nu }_4}\hfill \\ & & \left.+\left[{\partial }_{{\nu }_1}{\partial }_{{\nu }_2}{\partial }_{{\nu }_3}{\partial }_{{\nu }_4}{\varphi }^{\left(1\right)}\right]\right){\varphi }^{\left(2\right)}+\left({\tilde{t}}_1{{\varphi }_{0,0}^{\left(3\right)\nu \left(2\right)\nu }}_{\nu }-{\displaystyle \frac{2(t_0+t_1)}{d}}{\eta }^{{\nu }_1{\nu }_2}{{\varphi }_{0,0}^{\left(3\right)\mu \nu }}_{\mu \nu }\right)({\varphi }^{\left(1\right)}{\partial }_{{\nu }_1}{\partial }_{{\nu }_2}\hfill \\ & & -2\left[{\partial }_{{\nu }_1}{\varphi }^{\left(1\right)}\right]{\partial }_{{\nu }_2}+\left[{\partial }_{{\nu }_1}{\partial }_{{\nu }_2}{\varphi }^{\left(1\right)}\right]){\varphi }^{\left(2\right)}+{\tilde{t}}_2{{\varphi }_{0,0}^{\left(3\right){\nu }_1{\nu }_2}}_{{\nu }_1{\nu }_2}{\varphi }^{\left(1\right)}{\varphi }^{\left(2\right)}\},\hfill \end{array}$$

(141)

$$\begin{array}{ccc}& & \mathrm{where}{\tilde{t}}_1={\displaystyle \frac{t_1}{4}}-t_0,{\tilde{t}}_2\equiv -{\displaystyle \frac{1}{4}}\left(6t_1+{\displaystyle \frac{7t_2}{2}}\right).\hfill \end{array}$$

(142)

The errors do not influence the main results of the paper [1] and involves the interacting part of the action only in the tensor representation in ungauge form, as presented in the correct example in Appendix A.

8. In the Conclusions, the sentence, “An application of the gauge-fixing procedure admissible from the free formulations permits to present ⋯” should be shortened to the following:

An application of the gauge-fixing procedure admissible from the free formulations permits us to present the interacting Lagrangian, in both triplet tensor forms (132) and (A15), with off-shell traceless constraints (A16) and interacting action, depending on two sets of fields with irreducible deformed gauge transformations (137), (138). Then, in the tensor form (136), with only massless scalars ${\varphi }^{\left(i\right)}$, $i=1,2$, there is a basic massive field ${\varphi }^{\left(3\right)\nu \left(s\right)}$ and a set of auxiliary fields ${\varphi }_{0,2k}^{\left(3\right)\nu \left(s-2k\right)}$ in ungauge form for only a quartet of the unconstrained fields ${\varphi }^{\left(i\right)}\left(x\right)$, ${\varphi }^{\left(3\right)\nu \left(s\right)}$, ${\varphi }_{0,1}^{\left(3\right)\nu \left(s-3\right)}$, (139), (140), and (A33).

9. In Appendix A, the sentence after Equation (A20) with Equation (A21) should be changed as follows with adding of the Equations (A22)–(A32) (and with further reordering of formulae starting from (A22) according to the rule (AX + 10), for X = 22, 23, …)

Substituting in (A18) the fields ${|\varphi \rangle }_{s-r}$, $r\ge 4$ in terms of unrestricted quartet of fields $|{\varphi }_k{\rangle }_{s-k}$, $k=0,1,2,3$ from (A20), we obtain the action ${\mathcal{S}}_{C|s}^m\left(\varphi ,\dots \right){|}_{{|\varphi \rangle }_{s-r}={|\varphi \left({\varphi }_k\right)\rangle }_{s-r}}$ = ${\mathcal{S}}_s^m\left({\varphi }_0,{\varphi }_1,{\varphi }_2,{\varphi }_3\right)$, which is invariant with respect to the gauge transformations with two unrestricted gauge parameters:

$$\begin{array}{ccc}\hfill \delta |{\varphi }_0{\rangle }_s& =& l_1^{+}|{\xi }_0{\rangle }_{s-1},\delta |{\varphi }_2{\rangle }_{s-2}=l_1^{+}\left(2l_{11}\right)|{\xi }_0{{\rangle }_{s-1}+m|{\xi }_1\rangle }_{s-2},\hfill \end{array}$$

(A21)

$$\begin{array}{ccc}\hfill \delta |{\varphi }_1{\rangle }_{s-1}& =& l_1^{+}|{\xi }_1{\rangle }_{s-2}+m|{\xi }_0{\rangle }_{s-1},\delta |{\varphi }_3{\rangle }_{s-3}=l_1^{+}\left(2l_{11}\right)|{\xi }_1{{\rangle }_{s-2}+m\left(2l_{11}\right)|{\xi }_0\rangle }_{s-1}.\hfill \end{array}$$

(A22)

Instead, the gauge-fixing procedure for the tower of the gauge transformations for all fields and gauge parameters (starting from the first column in (A21), (A22))

$$\begin{array}{ccc}\hfill \delta |{\varphi }_2{\rangle }_{s-2}& =& l_1^{+}|{\xi }_2{{\rangle }_{s-3}+m|{\xi }_1\rangle }_{s-2},\hfill \end{array}$$

(A23)

$$\begin{array}{ccc}\hfill \delta |{\varphi }_3{\rangle }_{s-3}& =& l_1^{+}|{\xi }_3{\rangle }_{s-4}+m|{\xi }_2{\rangle }_{s-3},|{\xi }_2{{\rangle }_{s-3}=2l_{11}|{\xi }_0\rangle }_{s-1},\hfill \end{array}$$

(A24)

$$\begin{array}{ccc}\hfill \delta |{\varphi }_4{\rangle }_{s-4}& =& l_1^{+}|{\xi }_4{\rangle }_{s-5}+m|{\xi }_3{\rangle }_{s-4},|{\xi }_3{{\rangle }_{s-4}=2l_{11}|{\xi }_1\rangle }_{s-2}\hfill \\ \hfill \dots & & \dots \dots \dots \dots \dots \dots \hfill \end{array}$$

(A25)

$$\begin{array}{ccc}\hfill \delta |{\varphi }_{s-1}{\rangle }_1& =& l_1^{+}|{\xi }_{s-1}{\rangle }_0+m|{\xi }_{s-2}{\rangle }_1,|{\xi }_{s-2}{{\rangle }_1=2l_{11}|{\xi }_{s-4}\rangle }_3,\hfill \end{array}$$

(A26)

$$\begin{array}{ccc}\hfill \delta |{\varphi }_s{\rangle }_0& =& m|{\xi }_{s-1}{\rangle }_0,|{\xi }_{s-1}{{\rangle }_0=2l_{11}|{\xi }_{s-3}\rangle }_2\hfill \end{array}$$

(A27)

implies from (A27), (A26), the removing of the fields $|{\varphi }_s{\rangle }_0$, $|{\varphi }_{s-1}{\rangle }_1$ by means of use of $|{\xi }_{s-1}{\rangle }_0$, $|{\xi }_{s-2}{\rangle }_1$, so that the next gauge parameters $|{\xi }_{s-m}{\rangle }_{m-1}$, $m=3,4$ become traceless. Then, proceeding to (A22), we successively remove all traceless parts from the auxiliary fields $|{\varphi }_k{\rangle }_{s-k}$, $k\ge 1$ by means of the respective use of traceless $|{\xi }_{k-1}{\rangle }_{s-k}$ for $k=s-2,\dots ,1$. As a result, the system of the linear equations for the remaining ungauge fields in (A20) after changing the fields ${|\varphi \rangle }_{s-2k}$, ${|\varphi \rangle }_{s-2k-1}$ on their trace parts, except for $|{\varphi }_0\rangle$, takes the form

$$\begin{array}{ccc}\hfill & & {\left(l_{11}\right)}^{[s/2]-i{\theta }_s-1}|{\varphi }_{i+2}{\rangle }_{s-i-2}=2{\displaystyle \frac{([s/2]-i{\theta }_s-1)}{[s/2]-i{\theta }_s}}{\left(l_{11}\right)}^{[s/2]-i{\theta }_s}|{\varphi }_i{\rangle }_{s-i},i=0,1,\hfill \end{array}$$

(A28)

$$\begin{array}{ccc}\hfill & & l_{11}^{+}|\tilde{\varphi }{\rangle }_{s-2k-2-i}=k{\left(2l_{11}\right)}^{k-1}|{\varphi }_{2+i}{{\rangle }_{s-2-i}-(k-1){\left(2l_{11}\right)}^k|{\varphi }_i\rangle }_{s-i},k=1,\dots ,\left[\frac{s}{2}\right]-1-i{\theta }_s\hfill \end{array}$$

(A29)

(for ${\theta }_s\equiv {\delta }_{s-2[s/2],0}$). In (A28), (A29) and below, we use the decomposition of the fields $|{\varphi }_j\rangle$ into a sum of traceless fields $|{\varphi }_j^l\rangle$

$$|{\varphi }_j{\rangle }_{s-j}=|{\varphi }_j^0{\rangle }_{s-j}+\sum _{l=1}^{[s-j/2]}{\left(2l_{11}^{+}\right)}^l|{\varphi }_j^l{\rangle }_{s-2l-j}\equiv |{\varphi }_j^0{{\rangle }_{s-j}+l_{11}^{+}|{\tilde{\varphi }}_j\rangle }_{s-j-2}$$

(A30)

for $|{\varphi }_j^0{\rangle }_{s-j}=0$, $j\ge 1$). One can show that all fields $|{\varphi }_j^l{\rangle }_{s-2l-j}$ for $j\ge 2$; therefore, the total fields $|{\tilde{\varphi }}_j{\rangle }_{s-j-2}$ are expressed in terms of $|{\varphi }_0{\rangle }_s$, $|{\tilde{\varphi }}_1{\rangle }_{s-3}$ and their traces from the system (A28), (A29) for $k<:\left[{\displaystyle \frac{s-1-i}{2}}\right]\equiv k_{max}$

$$|{\varphi }_{2k+i}{\rangle }_{s-2k-i}=\sum _{l=1}^{k_{max}-k}{\alpha }_{k|l}^i{\left(2l_{11}^{+}\right)}^l{\left(2l_{11}\right)}^{k+l}|{\varphi }_i{\rangle }_{s-i},i=0,1,k\ge 1,$$

(A31)

with definite rationals ${\alpha }_{k|l}^0,{\alpha }_{k|l}^1$. Thus, the only s traceless fields, $[s/2]+1$ ones in $|{\varphi }_0{\rangle }_s$ and $\left[\right(s-1)/2]$ in $|{\tilde{\varphi }}_1{\rangle }_{s-3}$, compose the residual field vector

$$|\tilde{\Phi }{\rangle }_s=|{\varphi }_0{\rangle }_s+d^{+}{l}_{11}^{+}|{\tilde{\varphi }}_1{\rangle }_{s-3}+\sum _{i=0}^1\sum _{k=1}^{k_{max}}\sum _{l=1}^{k_{max}-k}{\displaystyle \frac{{\left(d^{+}\right)}^{2k+i}}{(2k+i)!}}{\alpha }_{k|l}^i{\left(2l_{11}^{+}\right)}^l{\left(2l_{11}\right)}^{k+l}|{\varphi }_i{\rangle }_{s-i}.$$

(A32)

The respective action will be equivalent to the Singh–Hagen action [46].

10. In Appendix A, the sentence after Equation (A22) (now enumerated as (A33)) should be changed to the following:

We would like to emphasize that there are a lot of equivalent Lagrangians with the same or different number of tensor fields in the configuration spaces. In this connection, the Singh–Hagen action can be considered as the only possible representative from the set of equivalent Lagrangians. The main requirement for any such a Lagrangian formulation for the given field with mass and spin consists in the fact that its dynamics must be in a one-to-one correspondence with one determined by the initial set of relations (1) on the physical field by selecting the element of irreducible Poincare group representation. Thus, we present in the appendix many equivalent Lagrangian formulations in various configuration spaces for free massive particles of spin s. None of them is more fundamental than the others. At present, there exists a general BRST formulation that allows us to derive any of them (see, e.g., the quartet formulation with a greater number of fields than in the Singh–Hagen formulation).