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the question of measuring of the spatial non - commutativity effects , in physical processes , is under intensive interest .
non - commutative qed ( ncqed ) seems to be a straightforward method to examine such effects . for this purpose
, one needs a precise experimental data such as positronium hyperfine splitting ( hfs ) among the other processes .
the basic difference between ncqed and qed is the existence of new interactions ( 3-photon and 4-photon vertices ) which complicate the calculations in ncqed .
although the feynman rules of this theory are given in @xcite , to apply these rules to bound state , one needs special treatments like bethe - salpeter ( bs ) approach @xcite or non - relativistic qed ( nrqed)@xcite . in our preceding letter @xcite , using bs equation , we have shown that up to the order @xmath4 no spin - dependent correction owing to the spatial non - commutativity appears in the positronium spectrum
. therefore one should calculate the higher order corrections . in this letter
we calculate the corrections to the positronium by using nrqed method . in section 2 , we introduce nrqed - vertices in the nc - space .
consequently , in section 3 , we use the modified nrqed to determine hfs at the lowest order . in this section , we show that our calculations at the leading order lead to the corrections at the order of @xmath0 , where @xmath1 is the parameter of non - commutativity . at the end , we summarize our results .
nrqed is an effective field theory which simplifies the bound state calculation . to apply this technique in non - commutative space one should modify the nrqed vertices by performing @xmath5 expansion on ncqed scattering amplitude . in doing so
, one obtains an effective theory of non - relativistic particles which permits the direct application of well tested techniques based on schrdinger s equation .
now , comparing ncqed scattering amplitudes with nrqed can completely determine the matching coefficients .
some of the vertices with their appropriate matching coefficients , are shown in fig .
1 . they contribute to the tree level matching to get the leading order bound state energy shift .
one should note that these coefficients apart from a phase factor are very similar to the standard nrqed @xcite .
this similarity is owing to the fact that the scattering amplitude of @xmath6 in ncqed is independent of the parameter of non - commutativity of space @xcite .
the other vertices which are not shown in fig . 1 and have not counterpart in the standard nrqed , due to the existence of the three and four photon vertices have contributions to higher order corrections to energy shift .
now , by using the first graph of fig . 1 and expanding the vertices up to order @xmath1 , one can easily verify the results of refs.@xcite at the order @xmath7 as @xmath8 such an energy shift is spin - independent and therefore has not any contribution to hfs .
by using the modified nrqed we can determine the diagrams which contribute to the lowest order of hfs ( fig .
2 ) .    we can now calculate each diagram separately as follows : @xmath9 with @xmath10_i \frac{-1}{({\bf p}-{\bf p}')^2}\nonumber\\ & & \left[\delta_{ij}-\frac{({\bf p}-{\bf p}')_i({\bf p}-{\bf p}')_j}{({\bf p}-{\bf p}')^2}\right ] \left[\frac{e({\bf
p}'+{\bf p})}{2m_e}e^{i p\wedge p'}\right]_j,\end{aligned}\ ] ] where @xmath11 and @xmath12 , the parameter of the non - commutativity is given as @xmath13.\end{aligned}\ ] ] it is shown that @xmath14 leads to some problems with the unitarity of field theory and the concept of causality @xcite , therefore in our calculations we consider @xmath15 . + after some algebra eq.([p1 ] ) yields @xmath16\psi({\bf r})\nonumber\\ & = & \frac{\alpha}{2m_e^2}\left<\frac{{\bf s}_1.{\bf l } } { r^3}\right>-\frac{3\alpha } { m_e^2}\int d^3 r ( \theta .{\bf l } \psi^*)\frac{{\bf s}_1.{\bf l}}{r^5}{\psi}+o(\alpha^7),\end{aligned}\ ] ] where @xmath17 and @xmath18 . in the third equality we used @xmath19
one should note that the first term in eq.([p4 ] ) is the usual term in nrqed which is of the order @xmath20 .
but the second term which is appeared in eq.([p4 ] ) , owing to the spatial non - commutativity is of the order @xmath0 .
nonexistence of the terms at the order of @xmath4 which carry @xmath1-dependence is a remarkable result which happens in all diagrams of hfs .
indeed this fact is due to appearance of @xmath21 instead of @xmath22 in all energy - correction expressions .
therefore , to obtain the energy corrections for hfs at the order @xmath23 one should once calculate the corrections up to the lowest order of nrqed ( i.e. fig .
2 ) . in the other words , the @xmath24-corrections in nrqed calculations of commutative space lead to the higher order of @xmath25 in non - commutative space .
+ now we work out the figs .
2(b - f ) as follows : @xmath26 @xmath27 with @xmath28_i \frac{-1}{({\bf p}-{\bf p}')^2}\nonumber\\ & & \left[\delta_{ij}-\frac{({\bf p}-{\bf p}')_i({\bf p}-{\bf p}')_j}{({\bf p}-{\bf p}')^2}\right ] \left[\frac{-ie({\bf p}'-{\bf p})\times{\bf{\sigma_2}}}{2m_e}e^{i p\wedge p'}\right]_j,\end{aligned}\ ] ] which results in @xmath29\frac{1}{4\pi r}\nonumber\\ & = & ( \ldots)+\frac{3e^2}{16\pi m_e^2}\int d^3r\psi({\bf r})\tilde\gamma_e\psi^*({\bf r}),\end{aligned}\ ] ] where @xmath30 means the usual part of the energy shift and @xmath31,\end{aligned}\ ] ] where @xmath32 . the final diagram ( fig .
2f ) has not any contribution at the order of our interest . for @xmath33
one can easily find @xmath34\ell\psi({\bf r})\nonumber\\ \triangle e^{\hbox{nc}}_a+\triangle e^{\hbox{nc}}_b&=&\frac{1}{2}(\triangle e^{\hbox{nc}}_c+ \triangle e^{\hbox{nc}}_d)\nonumber\\ \triangle e^{\hbox{nc}}_e&=&\frac{3e^2}{16\pi m_e^2}\int d^3 r \psi({\bf r})\overline{\gamma}\psi^*({\bf r})\end{aligned}\ ] ] where @xmath35 the superscript nc in eq .
( [ p11 ] ) means the non - commutative part of the energy shift and three lines in the eq.([p12 ] ) are related to @xmath36 , respectively .
meanwhile for the spin zero state ( @xmath37 ) , all contributions to the energy shift are zero .
+ the average of @xmath38 over the triplet is zero , which means the spin - spin interaction part carries no correction in average and therefore the hyperfine splitting due to the non - commutativity becomes @xmath39\ell\psi_{nlm}({\bf r}),\end{aligned}\ ] ] where @xmath40 is the wave function of the positronium in the commutative space with the coulomb potential and we have defined @xmath41 .
if the @xmath42-axis is chosen parallel to the vector @xmath43 , the above result simplifies into @xmath44 where @xmath45 is the compton wave length of the electron and @xmath46 is defined as @xmath47 one should note that the divergence of @xmath48 at @xmath49 is owing to singularity of @xmath50 at @xmath51 , the region where @xmath1-expansion is not well - defined .
actually , it is shown that @xmath1-expanded ncqed is not renormalizable @xcite .
the @xmath1 expansion imply a cut - off @xmath52 while the validity of nrqed requires @xmath53 .
since @xmath54 , the appropriate cut - off is @xmath55 .
therefore the energy shift for @xmath2 , @xmath49 can be obtained as @xmath56\alpha^6.\ ] ]    the above result should be added to the values of hfs derived in nrqed at the order @xmath23 .
the reported uncertainties on the experimental values of @xmath3 are about @xmath57 mhz @xcite , that give an upper bound @xmath58 . therefore determining the value of @xmath59
requires more accurate experiments .
using nrqed method in the non - commutative space , we have obtained that there is not any correction at the order @xmath4 for the hfs of positronium , the order @xmath4 corrections are spin independent .
the correction to the energy shift is started at the order @xmath23 , eqs .
( [ p13]-[p14 ] ) , and it depends on @xmath60 and @xmath61 quantum numbers . therefore it dosent have any contribution to the @xmath62 ( in the spectroscopic notation @xmath63 ) , while for @xmath64 there is @xmath65 different shifts .
consequently , a closer look at the spacing between @xmath2 triplet excited levels ( @xmath3 ) , which has already been measured [ 14 - 17 ] can provide an experimental test on the non - commutativity of space . + * acknowledgement * + s.m.z .
gratefully acknowledges research funding from shiraz university and ipm .
the research of m.h . and
f.l . was supported by isfahan university of technology ( iut ) and ipm .
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we obtain positronium hyperfine splitting owing to the non - commutativity of space and show that , in the leading order , it is proportional to @xmath0 where , @xmath1 is the parameter of non - commutativity . 
 it is also shown that spatial non - commutativity splits the spacing between @xmath2 triplet excited levels @xmath3 which provides an experimental test on the non - commutativity of space .    
 epsf.tex = -.1 cm = 16 cm = 24.5 cm = -2 cm = -2 cm