Прикладная физика и математика 2014 №1

APPLIED PHYSICS AND MATHEMATICS

1 2014

ISSN 2307-1621

ISSN 2307-1621 1 2014

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: 77-50415 25.06.2012 .

: 83190 10363

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1 2014

SCIENTIFIC JOURNALISSN 2307-1621 1 2014

AppLIEd phySICS ANd MAThEMATICS

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APPLIED PHYSICS

.. Chernobaeva, .. SkundinTHErmAL AgIng EffECTS on of Low-CArbon Low ALLowS STEELS mECHAnICAL ProPErTIES bEHAvIEr 3

A.I. MilanichDIrECT mEASurEmEnT of HubbLE ConSTAnT from DISTAnCE To THE moon 26

S.A. YuditskiyInformATIon-EnErgETIC InTErACTIonS AnD rESPIrATorY-TExTuAL mEDITATIon mETHoD 29

APPLIED mATHEmATICS

C. Daviau, J. BertrandnEw InSIgHTS In THE STAnDArD moDEL of quAnTum PHYSICS In CLIorD ALgEbrA (PArT 1) 35

I.I. Bugaets, I.V. NekrasovCLASSICAL LAgrAngE ProbLEm-bASED SYnTHESIS of ProgrAm-CorrECTED ConTroL STrATEgY for A non-LInEAr DYnAmIC SYSTEm 66

ruLES of ConSIDErATIon, PubLICATIon AnD rEvIEw ArTICLES 74

Content

1 2014 3

.. . .. , E-mail: [email protected]

, , - . , - . - 152. - -

152 - . -, , , .

: , -, , , .

.. ChERNoBAEVA Doctor of Tech. Sciences .. SkUNDIN nrC Kurchatov Institute moscow, russian federation E-mail: [email protected]

THErmAL AgIng EffECTS on of Low-CArbon Low ALLowS STEELS mECHAnICAL ProPErTIES bEHAvIEr

The paper presents reviews not numerous papers, concerned the study of thermal aging effects in forging from low-carbon low allows steels. The analysis of data, contained in reviewed papers, has been done to reveal the mechanisms of mechanical properties behavior under thermal aging. Special attention has been paid to data analysis of thermal aging of steel 15Kh2NMFA. Study of surveillance specimens from steel 15Kh2NMFA shown

no reasons for hardening changing. The reviewed data shown, that use of Hollomon equation for the recalculation of data from higher temperature thermal aging to lower temperature thermal aging can give not correct assessment of effects.

Key words: Thermal aging, yield stress, transition temperature, hardening, embrittlement.

- [1, 2]. , - , [2]. - , , .

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1 2014 1 20144

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- . , - , - - . - , -, - , , - [1].

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, - . , -, . [325] - , - .

[5] 10 - , - 0 15000 340450 . - :1)

:

(tk = const) = 0exp(Q / (RT )), (1)

R , 0 , Q .2) Q -

. -

. = 3040 - 110117 /, = 5080 196226 /. 122 [7]. - [5] Q - , - .

7183,5 / [26, 27]. - 217 /, Fe-1,3% Cr 234 /, Fe-0,47 % Mo 258 / [28]. [28] - Q 155192 /. = 50 Q - , [5, 7]. , -, , . , .

[57] - , - . [8, 29, 30] . [10, 31] - .

- -, , .

[36, 2325] 10 101. 1 - .

1 - - , ,

1 2014 5

(, ). - 10 (~7%) [5] (4550%) (5055%). - 101 . -, ( 15000 ) 340450 (0,2 ) (, ) 10 101 ( 2) [5, 6].

, 2, , 101 340,

400 15000 - . 450 - 100 .

10 - 400 450 - , 340 100 .

340, 400 450 - -. 1 10.

1. 10 101 [36]

C Mn Si Cr Ni Cu S P

10 900-990 ; 680-690 0,09 0,63 0,79 0,65 0,50 0,42 0,020 0,020

101 900-920 ; 670-690 0,09 0,45 0,28 0,62 1,72 0,20 0,009 0,010

2. 10 101 [5, 6]

( , %)

0,2, /2 , /2 t..,

T, , .

10

514 634 -40

340 15000 621 751 10

400 15000 534 678 80

450 15000 542 587 100

101

531 623 0

340 15000 495 611 10

400 15000 530 629 20

450 15000 420 494 20

. 2 (1,3) - (5), (2, 4) 400 (1, 2) 450 (3-5) - 10 [3]

. 1 10 - [5]

1 2014 1 20146

1 , , . , - ( 1000 ). , - ~100 450 15000 .

10 , [3]:1) -

10 (- 2).

2) , - .

3) , . [20]

(1000 ) 450 10. - 920 680 1,5 .

[20], - - (0,2 ): 0,66% . 0,2 493

625 , - 0,88 % . 0,2 740 796 . 0,020 0,043% - : 0,2 20. - . , 50 % , - 450 1000 160 .

10 - - , - . (~109 -2). -- [23] . , .

450 1000 . -. . - - 1426 ( 3).

3. 10

d, d.(), dc(), Nv,

-3 Nv(), -3 d, Nv(), -3

1

0,660,07 0,110,07 0,067 14,17 5,69 0,026 33,220,3

450 /1000 0,620,14 0,090,05 0,066 24,20 11,03 0,018 15832

2

0,630,20 0,080,06 0,061 22,07 14,60 0,016

450 /1000 0,560,11 0,070,04 0,050 27,60 11,60 0,014 21335

3

0,790,24 0,070,05 0,050 22,90 10,7 0,017

450 /1000 0,520,12 0,090,12 0,060 0,014 13530

4

0,670,17 0,100,07 0,060 18,43 5,89 0,024 3911

450 /1000 0,600,25 0,090,05 0,060 22,5 10,63 0,023 4912

d , d.() , dc() , Nv , Nv() , d , Nv()

1 2014 7

- 10 [4]. 0,008 0,035 % . 3 . - [25], 10 0,035 % . 180 450 .

, - , , - 3, - .

[6] - 101. 4 101 340, 400 450 .

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- [36, 2325]

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3) .

4) Fe3C - 340450 . , ,

, - , - [6].

[25] 101. 0,010 0,044 % . . 5 400 450 . , - . - : - .

. 3 t... 400 (1, 3) 450 (2, 4) 10 0,008% (1, 2) 0,035 % (3, 4) [4]

. 4 t... - 340(1), 400(2) 450(3) 101 [6]

1 2014 1 20148

101 , 0,010 % 0,044% , , - - (2,40,9 ) (3,61,1 ). - , - 4,90,9 . - , -

- [25].

- 10 [4]. 0,008 0,035 % . 3 - . [25]. 0,008 0,035 % 10 4 450 .

10 101 - , .

, -, -, 10 - ( 3, 5).

[7] 122. 4.

, - , -: 920 670690 1,5 .

4. 122 [7] , %

C Mn Si Cr Ni Mo V Al S P

0,18 1,01 0,50 0,48 1,47 0,19 0,12 0,09 0,021 0,007

. 5 t... 101 0,010% (1, 2) 0,044% (3, 4) - 400(1, 3) 450 (2, 4) [25]

. 6 0,2, 122 340(1) 550(2) [7]

1 2014 9

. 10 101 122 - 340550 ( 6).

, 6, - ~1000 - . - - , 0,17 - .

7 [7] - - 122.

7, - 500 - 450 . 10000 400 - . - 450550 500 1000 - ( ) 2030 .

- [7] ( 8). - . - 400500 6080 . 550 - . - [7] .

[7] 122 - , 10 101: 350450 : 10000 . - - . - [7] , 122 - - , - . ~1010 2. (0,10,3 ) - . - . , - 122 340450 . (500550 ) (58109 2) . [7] - 1000 , - . 0,2 - , , - .

, , 8, , - , 550 500 , .

, 101, 10 122 - - 350550 :

. 7 0,2, , - 500(1) 10000 .(2) - 122 [7]

. 8 t... 122 - 340(1), 400(2), 450(3), 500(4), 550(5) [7]

1 2014 1 201410

, - , .

.

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- [9] -533. 5.

- -533 900 , - , 671 621 .

- - ( 20000 .) - 300, 400, 450, 500 550 . 9 - .

-

5. -533 [9]

, %

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0,22 0,017 0,63 0,20 1,40 0,19 0,008 0,54 0,19

. 9 -533 - -; [9]

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1 2014 11

.

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(- 10) 2050Hv - 330 365 . 405, 450 500 . - 330 20000 . (~15 Hv).

. 10 -533 ; , . , Hv [10]

. 11 -533 - 365; , . , Hv [10]

1 2014 1 201412

6. , [31]

Cu Ni Mn Si P C Mo Cr

, % 0,44 1,66 1,38 0,75 0,018 0,19 0,24 0,054

. 12 -533 - 365; , . , Hv [10]

500 - . 11, , - (~50Hv ~20Hv ).

, , - 10 11, - , .

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1 2014 13

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, [10, 31] , - - . -, - - .

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- : 940960 , 660680 2324 . 350, 400, 450, 500 550 - 30000 ( 14).

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, %

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0,17 0,37 0,27 2,96 0,75 0,67 0,28 0,07 0,008 0,007

1 2014 1 201414

4000 350 . - - 350 .

14, 152- . 350 . 30 ~ 40000 .

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-1000, , 152. 152-. - 5 , 1 . - 152.

-1000 . - , . - - 900920 650 . ( ). ~290 , ~320 .

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Si Mn Cr Ni Mo V P S Cu

152 108-765-78

min 0.13 0.17 0.3 1.8 1 0.5 0.1

max 0.18 0.37 0.6 2.3 1.5 0.7 0.12 0.02 0.02 0.3

152- 108-765-78

min 0.13 0.17 0.3 1.8 1 0.5 0.1

max 0.18 0.37 0.6 2.3 1.5 0.7 0.12 0.01 0.01 0.08

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17 .

, 17, , - . 0,27 .

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. 18 0,2 t50 350 152- (1) 102 (2) [29]

. 17 152- 350 [35]

1 2014 17

- [29].

-. - . - - [29]. - , 102 - - 152-: 102 1,5109 2, 152- 4,5109 2 [29].

- : , - - , , - , , , , . [36] - [29] 102 - M3C, M2C, M23C6, VC, 152- M3C, M7C3, M2C, VC, M23C6.

350 10000 102 - , - ( 19114 1078 ) - . - . 152- - - , ( 1517 813 1096 . , , , 152- 350 - , - [29].

[38] , - , -, . 152- 1 720 , 102 690 , - -.

102 152-. - - 102.

152-, 102, - 152-. 152- , . - [30, 33, 37, 38].

[30] - 350 - 152- 102. 19 , .

, - 10003000 2030 . 10 300010000 . - 300 - 7000 . , , 350 300 , 40008000 , [30].

, - . - . 250, 300, 350 400 1000, 3000, 5000, 7000 10000 . t~(1/) ( 21). t~(1/) , - (1).

[30] , - , , [38]. - : , - , 50000 , - . - 30 . :

1 2014 1 201418

= (max) (1 -b(t-t(max))), (2)

(max) ,, b , t(max) , - .

(1.2) 2,3 . 21 22 -

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, [35]: , - 30 , , .

, 152 350 - 10000 - , . - .

90- --. , -. , - - .

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. 21 - 152- 20(////) 350(\\\\), , - .-08 20() 350() [37]

. 20 t 1/, =const ( ): , 152-; , 102 [30]

1 2014 19

. t, D 20 . , 50000 - (~320 ) D=20 ( 23). .

- , - 20000 120000 , ( 24).

[38] -. ( 26):

+=

198003267040740

exp2,26134,2)(tthttTT (3)

D -, t ,

- - :1) ; 2)

- ;

3) , - . , . -

300, , - , . [40] - 320 :

+=

198001980024600

exp)(T infTtthtbTt t (4)

inf t;bt , ;

26 - ( (1.4); bt = 13,9, Tinf = 2,7) - .

- -, 0,26 . , (4)

. 22 152- (); 152- - (b): =300 (1) =0 (2) [37]

. 23 - , - -- [39]

1 2014 1 201420

-- .

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, - . , :

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: - - - V(C, N) Cr23C6.

. 25 - 300 () Ni < 1,3% () Ni > 1,5% ()

. 26 (4),

. 24 , - -; ) ) [39]

1 2014 21

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- - ~20 . -- . , - .

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. 28

( 27) . -, [4143] , - . - , .

1 2014 1 201422

9. -1000

-

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, 50020000 10000170000 10000170000

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* ** 0

, 29, - , . - - - . - , , - .

-

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. 29 -

. 30 0,2 -

1 2014 23

- () - - 0 . - 0 , .

.

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, - - , :

[38] - - 350. - ;

, - - ( );

- [59] - - ;

, , - - 350550 , ;

, - , , - , - ;

- , - - .

1. .. , .. , .. . - . . -: , 1999.

2. .. . . . 3-. ., , 1978.

3. .. , .. , .. , .. -. 10 340450 // . 1976. 7. . 6573.

4. .. , .. , .. , .. -. - 10 / .: -. . . . 28. .: , 1975. . 3339.

5. .. , .. , .. . / - 10 // .: . . . 21. .: , 1975. . 5663.

6. .. , .. , .. . - - 101 // . 1978. 2. . 9399.

7. .. , .. , .. , .. . - 122 // . 1978. 2. . 9399.

8. .. , .. . - 0,60,8% // - . 2009. 2. . 58.

9. Materials Reliability Program: A Review of Thermal Aging Embrittlement in Pressurized Water Reactors (MRP-80), EPRI, Palo Alto, CA: 2003. 1003523.

10. Keith Wilford, Dave Ellis, Tim Williams, S. Hirosawa, G. Sha, A. Morley, A. Cerezo, G.D.W. Smith. Thermal Ageing in RPV Steels // IGRDM-14, Pittsburgh, PA, USA, 2125 April 2008

11. .. . - - : . . . / . . . (). ., 1971.

12. .. . . .: , 1949. 592 .

13. .. , .. , .. . - , - , . - , 1976, . 41, . 1. . 99111.

14. .. . - - : . . . / . .-. (). ., 1972.

15. .. , .. . - - . .: , , - . ., 1976, . 154159.

16. T. Mukherjee. Kinetics of coorsening of carbides in chromium steels at 700 . J. Iron and steel Inst., 1969, Vol. 207, 5, . 621631.

17. .. , .. . . -. 1970, 3, . 6668.

1 2014 1 201424

18. A.B. Muhammad, Z.C. Skopiak, M.B. Woldron. Effect of types of carbides on temper embrittlement in commercial 2.25Cr 1Mo steel. Adv. phys. met. and appl. steels. Proc. int. conf., Liverpool, 2124 sept., 1981. London, 1982, . 340348.

19. B.C. Edwards, B.L. Eyre, G. Gage. Temper embrittlement of low alloy Ni-Cr steels 1. The susceptibility to temper embrittlement and the influence og intermediate tempering treatments. Acta met., 1980, Vol. 28, 3, . 335356.

20. Y. Otoguro, T. Zaizen. Intergranular embrittelment in steels. J. Ma-ter. Sci. soc. Jap., 1980, Vol.17, 2, . 8896.

21. T.C. Lei, C.H. Tang, M. Su. A new mechanism of the high tem-perature (reversible) temper brittleness of alloy steels. 3-rd IFHI Int. congr. Heat treat. Mater., Changhai, 711 Nov., 1983. London Chameleon press, 1983, 2/12/10.

22. .. . . , 1980, 155. . 5760.

23. .. , .. , .. . - - // . . . 1982. 2. . 150156

24. .. , .. , .. . , 10 // . 1979. 9. . 1618.

25. .. , .. , .. . - 101 // . 1981. 12. . 2931.

26. .. , .. . : 2- ., . . .: . 1980. 493 .

27. .. , .. , .. , .. . // .: . .: , 1964, . 305310.

28. .. , .. . // .: - . .:, 1964, . 311320.

29. .. , .. , .. , .. . // . 1992. 1. . 26.

30. .. , .. . 60 . //, 2006, 7. . 2327.

31. P.D. Styman, J.M. Hyde, A. Morley, K. Wilford & G.D.W. Smith, Precipitation in long term thermally aged high copper, high nickel model RPV steel welds, IGRDM 16, Santa Barbara, 2011.

32. .. , .. , .. , .. . - // . 1984. 10. . 5760.

33. .. , .. . -1000. // ISSN 01311336. . 12. 2009.

34. .. . - - . / . , 1982. . 128139.

35. .., .., - Cr-Ni-Mo Mn-Ni-Mo -, 177. . , 1988.

36. .. . - // . 1986. 11. . 2125

37. I. V. Gorynin, B.T. Timofeev. Aging of materials of the equipment of nuclear power plants after designed service life. // Materials Sci-ence, Vol. 42. 42. 2006 . 155169.

38. .. , .. , .. , .. , .. . - -1000 // ISSN 19946716. , 4. 2009. . 108123.

39. Nikolaev Yu.A., Radiation embrittlement of Cr-Ni-Mo and Cr-Mo RPV steels, journal of ASTM International, vol. 4, No. 8, 2007.

40. 1.3.2.01.00612009 -1000 -

41. .., .., .., .., .., - , . , 2012.

42. Gurovich .., .., .., .., .. (2012) - - - 1000 , 7, . 2225.

43. Gurovich .., .., .., .., .., .., .. (20121) - 152 - . . - , 4, . 110122.

44. A.A. Chernobaeva, E.A. Kuleshova, M.A. Skundin, D.A. Malsev, L.I. Chyrko, V.N. Revka, VVER-1000 base metal thermal aging surveillance specimens data base revision, proceeding of SMiRT-22 conference, USA, 2013.

45. .. -1000 - , , - , 2013.

References1. M.. Smirnov, V.. Schastlivtsev, L.G. Schastlivtsev. Basis of steel

heat treatment. Textbook. Ekaterinburg: Uro RAN, 1999.2. I.I. Novikov. The theory of metal heat treatment. Textbook. 3-d Edi-

tion. ., Metallurgiya, 1978.3. V.I. Bogdanov, S.. Vladimirov, L.I. Gladshteyn, V.. Goritsky.

Thermal aging of low alloy steel 10KhSND during long hold at 340450 // Problemy prochnosty. 1976. 7. . 6573.

4. V.I. Bogdanov, L.I. Gladshteyn, V.. Goritsky, Yu.I.Zvezdin. Effect of phosphorus on thermal aging and fractografy feature of specimens from steel 10KhSND / In book.: Voprosy sudostroeniya Ser. Metal science. Vol. 28. L.: ZNII Rumb, 1975. . 3339.

5. V.I. Bogdanov, S.. Vladimirov, L.I. Gladshteyn et all. / Study of structure evolution in steel 10KhSND during high temperature long term aging // In book.: Voprosy sudostroeniya Ser. Metal science. Vol. 21. L.: ZNII Rumb, 1975. . 5663.

6. V.I. Bogdanov, L.I. Gladshteyn, V.. Goritsky. Structure and ther-mal aging embrittlement resistance low alloys steel 10KhN1M // Problemy prochnosty. 1978. 2. . 9399.

7. V.. Goritsky, L.I. Gladshteyn, V.N. Orlova, E.I. Pichiy. Tendency to thermal aging embrittlement low alloys steel steel 12GN2FYu // Problemy prochnosty. 1978. 2. . 9399.

8. .. Zotova, I.V. Teplukhina. Study of thermal aging effects on em-bryttlement of RPV steel with Ni content 0,60,8% // Voprosy mate-rialovedeniay. 2009. 2. p.58.

9. Materials Reliability Program: A Review of Thermal Aging Embrit-tlement in Pressurized Water Reactors (MRP-80), EPRI, Palo Alto, CA: 2003. 1003523.

10. Keith Wilford, Dave Ellis, Tim Williams, S. Hirosawa, G. Sha, A. Morley, A. Cerezo, G.D.W. Smith. Thermal Ageing in RPV Steels // IGRDM-14, Pittsburgh, PA, USA, 2125 April 2008.

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11. S.S. Lvova. Study of alloying and thermal treatment effects on struc-ture and properties of steels for high power rotors of steam turbine and generators: Avtoreferat kand. dissert. ZNIITMASH. ., 1971.

12. Ya.S.Umansky. Physics base of metal science. .: Metalurgizdat, 1949. 592 p.

13. Yu.I. Ustinovshikov, I.. Kovensky, V.. Vlasov. - , , . Physics of Metal and Metal Science, 1976, Vol. 41, ser. 1, . 99111.

14. V.S. Sheyko. Development of chemical composition high strangest steel and heat treatment for welded rotors of high power turbines: Avtoref. kand. dissert. ZNIITMASH ., 1972.

15. P.. Antikayn, Yu.P.Bulanov. Changing of carbide phase and kinetic of carbide desolation at recovery heat treatment perlitic steels. In book.: Diffusion, phase transformations and mechanical properties of metal and alloys. ., 1976, . 154159.

16. T.Mukherjee. Kinetics of coarsening of carbides in chromium steels at 700 . J. Iron and steel Inst., 1969, Vol. 207, 5, . 621631

17. .Z. Kontorovsky, .. Rivlin. Structure changings at long term thermal aging low alloys turbine steels. Metal scince and heat treat-ment of metals. 1970. 3. . 6668.

18. A.B. Muhammad, Z.C. Skopiak, M.B. Woldron. Effect of types of carbides on temper embrittlement in commercial 2.25Cr 1Mo steel. Adv. phys. met. and appl. steels. Proc. int. conf., Liverpool, 2124 sept., 1981. London, 1982. . 340348.

19. B.C. Edwards, B.L. Eyre, G.Gage. Temper embrittlement of low alloy Ni-Cr steels 1. The susceptibility to temper embrittlement and the influence og intermediate tempering treatments. Acta met., 1980, Vol.28, 3. . 335356.

20. Y. Otoguro, T. Zaizen. Intergranular embrittelment in steels. J. Ma-ter. Sci. soc. Jap., 1980, Vol.17. 2. . 8896.

21. T.C. Lei, C.H. Tang, M.Su. A new mechanism of the high tempera-ture (reversible) temper brittleness of alloy steels. 3-rd IFHI Int. congr. Heat treat. Mater., Changhai, 711 Nov., 1983. London Cha-meleon press, 1983, 2/12/10.

22. V.. Yukhanov. Effect of long term thermal agings on structure and properties of RPV perlitic steel. Trudy ZNIITMASH, 1980. 155. . 5760.

23. V.. Goritsky, L.I. Gladshteyn, G.R. Shneyderov. Study of thermal embrittlement of low alloy steels. Izv. N USSR. Metals. 1982. 2. . 150156.

24. V.I. Bogdanov, L.I. Gladshtain, V.. Goritsky. Effect of phosphorus, vanadium and molybdenum on thermal aging effects of steel 10Kh-SND. Mi. 1979. 9. . 1618

25. V.I. Bogdanov, V.. Goritsky, Yu.I.Zvezdin. Effect of phosphorus on on thermal aging behavior and fracture features of steel 10KhN1. i. 1981. 12. . 2931.

26. Yu.. Lakhtin, V.P. Leontyeva. Metal science: Text book for ma-chine-building institutes 2-d publication., corrected. .: Mashi-nostroenie. 1980. 493 p.

27. V.. Borisov, V.. Golicov, .S. Savilov, V.. Sherbitsky et al. The study of diffusion of carbon in Fe. In book.: Tasks in metal science and metal physics. .: tallurgiya, 1964, . 305310.

28. P.L. Gruzdin, V.V. Mural. Study of phosphorus diffusion in Fe and its allows by radiometric method. In book.: Tasks in metal science and metal physics. .: tallurgiya, 1964. . 311320.

29. V.. Goritsky, G.R. Shneyderov, .D. Shur, V.. Yukhanov. Struc-ture mechanism of thermal aging in steels with structure of tempered bainite. i. 1992. 1. . 26.

30. V.. Yukhanov, .D. Shur. Study of thermal aging of RPV steels for NPP to base its lifetime till 60 years. i, 2006, 7 .2327

31. P.D. Styman, J.M. Hyde, A.Morley, K.Wilford & G.D.W. Smith, Precipitation in long term thermally aged high copper, high nickel model RPV steel welds, IGRDM 16, Santa Barbara, 2011

32. .. Astafyev, V.. Yukhanov, .D. Shur, V.V. Bobkov. Method of RPV steels because of thermal aging effects. Zavodskaya laborato-riya. 1984. 10. . 5760.

33. .V. Dub, V.. Yukhanov. VVER-1000 RPVs lifetime assessment. // ISSN 01311336. Tyazeloe mashinostroenie. 12. 2009.

34. V.. Goritsky. Effect of microstructure feature on fracture toughness under thermal aging in building steel. Trudy / ZNIIProektstalkon-strukziya. Moscow, 1982, .128139.

35. .. Sandomirsky, V.P. Savukov., Effect of heat treatment on tran-sition temperature and tempering resistense of Cr-Ni-Mo and Mn-Ni-Mo steels, Trudy NPO ZNIITMASH 177, . , 1988

36. .. Fstafyev. Tendency to embrittlement of Cr-Ni construction steels // i. 1986. 11. . 2125.

37. I. V. Gorynin, B.T. Timofeev. Aging of materials of the equipment of nuclear power plants after designed service life. // Materials Science, Vol. 42 42. 2006 .155169.

38. B.Z. Margolin, V.. Nikolaev, .V. Yurchenko, Yu..Nikolaev, D.Yu.Erak. Analysis of VVER-1000 RPV materials embrittlement under operation conditions. ISSN 19946716. Voprosy materialove-deniya, 4. 2009. . 108123.

39. Yu.A.Nikolaev, Radiation embrittlement of Cr-Ni-Mo and Cr-Mo RPV steels, journal of ASTM International, vol. 4, No. 8, 2007.

40. 1.3.2.01.00612009 Polojenie po kontrolyu mekhanicheskikh svoistv metalla ekspluatiruyushisa korpusov reaktorov VVER-1000 po resultatam ispytaniy obraszov svideteley.

41. B.A. Gurovich, E.A. Kuleshova, D.A. Maltsev, S.V. Fedotova, A.S. Frolov, Connection between operation characteristics of RPV steels and nanostructure evolution under operation temperature and irradiation, Alushta, 2012.

42. B.A. Gurovich, E.A. Kuleshova, S.V. Fedotova, A.S. Frolov, D.A. Maltsev. (2012) Phase transformation in surveillance speci-mens materials under thermal aging at VVER-1000 RPV operation temperature. Tyazeloe mashinostroenie. 7. . 2225.

43. B.A. Gurovich, E.A. Kuleshova, D.A. Maltsev, S.V. Fedotova, A.S. Frolov, .. Zabusov, .. Saltykov (20121) Study of struc-ture evolution of steel 15Kh2NF and its welding seam after long term thermal aging at RPV operation temperature. Izvestya vu-zov. Yadernya energetika. 4. . 110122.

44. A.A. Chernobaeva, E.A. Kuleshova, M.A. Skundin, D.A. Malsev, L.I. Chyrko, V.N. Revka, VVER-1000 base metal thermal aging surveillance specimens data base revision, proceeding of SMiRT-22 conference, USA. 2013.

45. .. Skundin VVER-1000 RPV Mechanical properties behavior under long term thermal aging at operation temperature., Avtore-ferat dissertazii na soiskanie stepeny kandidata tekhnicheskikh nauk, NRC Kurchatov Institute, 2013.

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123182, ,

, 1-mail: [email protected]

Chernobaeva Anna AndreevnaDoctor of Tech. Sciences, Leading ResearcherSkundin matvey AleksandrovichJunior ResearcherNRC Kurchatov Institute123182, Moscow, Russian FederationAkademika Kurchatova pl., 1-mail: [email protected]

Information about the author

1 2014 1 201426

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A.I. MIlANICh Doctor of Tech. Sciences, Professor, Head of Laboratory Institute of general Physics russian Academy of Science moscow Institute of Physics and Technology (mIPT) Dolgoprudnyj, russian federation, E-mail: [email protected]

DIrECT mEASurEmEnT of HubbLE ConSTAnT from DISTAnCE To THE moon

A new model for Moon Earth distance enlarging was discussed. In the range of experimental accuracy for lasers distance measure-ments this enlarging corresponds to expanding Universe with the Hubble constant 96 km/sMpc (modern value is 67 km/sMpc). It is the first direct estimation of the Hubbles constant alternative to

redshift measurements. So, based on this model we dont need tidal effects for explanation Moons orbital motion. Also this model and results are a strong argument against tired light hypothesis.

Keywords: Moon, ranging, tidal effects, Hubble constant, laser.

1 2014 27

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5. .., .., .. , -, // . 1999. . 169. 10. . 11451146.

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References1. Milanich A.I. Kvant prostranstva [Quantum of space]. Trudy 56

nauchnoj konferencii, Obshhaja i Prikladnaja Fizika [Proceed-ings of 56 scientific conference, General and Applied Physics]. Moscow Dolgoprudnyj Zhukovskij. 2013. P. 115117.

2. Aleshkina E.J. Lazernaja lokacija Luny [Lunar laser ranging]. Priroda [Nature]. 2002. 9. P. 5766.

3. Kokurin J.L. Lazernaja lokacija Luny. 40 let issledovanij [Lu-nar laser ranging. 40 years of research]. Kvantovaja jelektron-ika [Quantum Electronics]. 2003. Vol. 33. 1. P. 4547.

4. Ade P.A.R. et al. (Planck Collaboration) (2013). Planck 2013 results. I. Overview of products and scientific Results. arXiv:1303.5062.

5. Okun' L.B., Selivanov K.G., Telegdi V.L. Gravitacija, fotony, chasy [Gravitation, photons, clock]. UFN [Physics-Uspekhi]. 1999. Vol. 169. 10. P. 11451146.

6. Zel'dovich Ja.V., Novikov I.D. Odnoznachno li objasnenie krasnogo smeshhenija rasshireniem Vselennoj? [Definitely

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an explanation redshift expansion of the universe?]. Stroenie i jevoljucija Vselennoj [Structure and Evolution of the Uni-verse.]. M.: Glavnaja redakcija fiziko-matematicheskoj liter-

atury izdatel'stva Nauka [Moscow: Main Edition physical and mathematical literature publishing house Science], 1975. P. 123124. 736 .

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milanich Alexander IvanovichDoctor of Tech. Sciences, Professor, Head of Laboratory Institute of General Physics Russian Academy of Science119991, Moscow, Russian Federation, Vavilova str., 38 Moscow Institute of Physics and Technology (MIPT) 141707 Dolgoprudnyj, Russian FederationInstitutskiy line, 9, E-mail: [email protected]


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S.A. YUDItSkIY Doctor of Techn. Sciences, Professor, Senior researcher v.A. Trapeznikov Institute of Control Sciences of russian Academy of Sciences moscow, russian federation

InformATIon-EnErgETIC InTErACTIonS AnD rESPIrATorY-TExTuAL mEDITATIon mETHoD

Here is summarized information-energetic interactions concept of human psychic sphere and Forces of Nature. Within the concept explained and described a new meditation technique based on diaphragmatic breathing synchronization and mental utterance of rhythmic poetic texts. Proposed technology should be considered as a psychological method, aimed at

forming positive emotional attraction and, as a consequence, a positive circle of human thoughts.

Keywords: information-energetic field, fields interaction, respi-ratory-textual meditation, diaphragmatic breathing, poetic text, text markup, breath and text synchronization.

1 2014 1 201430

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References1. Vajnberg S. Mechty ob okonchatel'noj teorii: fizika v

poiskah samyh fundamental'nyh zakonov prirody: per. s angl. [Dreams of a Final Theory: The Scientists Search for the Ultimate Laws of Nature]. M.: Editorial URSS /URSS, 2004. 256 p.

2. Grin B. `Elegantnaya Vselennaya. Superstruny, skry-tye razmernosti i poiski okonchatel'noj teorii: per. s angl. [The Elegant Universe: Superstrings, Hidden Di-mensions, and the Quest for the Ultimate Theory]. M.: Librokom [Moscow: Publishing house Librokom], 2011. 288 p.

3. Uolsh N. Besedy s Bogom: per. s angl. [The Complete Conversations With God] K.: Sofiya, M.: Gelios. Kn. 1, 2001, kn. 2, 2002, kn. 3, 2002.

4. Amen D. Mozg i dusha. Novye otkrytiya o vliyanii moz-ga na harakter, chuvstva, `emocii: per. s angl. [Healing the Hardware of the Soul] M.: Eksmo. [Moscow: Pub-lishing house Eksmo], 2012, 304 p.

5. Hej L. Isceli sebya sam: per. s angl. [Heal your body]. M.: Olma-Press [Moscow: Publishing house Olma-Press], 2006. 208 p.

6. Sinelnikov V.V. Vozlyubi bolezn svoyu [Thou shalt love your sickness]. M.: Centrpoligraf,M.: Centrpoligraf [Moscow: Publishing house Centrpoligraf], 2009. 415 p.

7. Konovalov S. S. Chelovek i Vselennaya [Human and Universe]. SPb.: Praym-Evroznak, 2010, 192 p.

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117997, , . , . 65

S.A. YuditskiyDoctor of Techn. Sciences, Professor, Senior Researcher V.A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences117997, Moscow, Russian FederationProfsoyuznaya street, 65


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C. Daviau Le Moulin de la Lande 44522, Pouille-les-coteaux France, mail: [email protected]. BertranD 15 avenue Danielle Casanova 95210, Saint-Gratien France, mail: [email protected]

New iNSiGhtS iN the StaNDarD MoDeL oF quaNtuMPhySiCS iN CLiorD aLGebra (Part 1)

Abstract : This is the rst of three parts. It presents the needed Cliord algebras. We explain how the form invariance of the Dirac equation implies the use of the space algebra. We present a nonlinear wave equation which has the Dirac equation as lin-ear approximation. We extend the invariance group, both to the

quantum wave and to the electromagnetism with photons and magnetic monopoles.

Keywords : invariance group, Dirac equation, electromagne-tism, Cliord algebras, magnetic monopoles.

. 44522, , . mail: [email protected] . 95210, , -. 15 - mail: [email protected]

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introductionTo see from where comes the standard model that rules today quantum physics, we first return to its beginning. When the idea of a wave associated to the move of a par-ticle was found, Louis de Broglie was following conse-quences of the restricted relativity [14]. The first wave equation found by Schrdinger [30] was not relativistic, and could not be the true wave equation. In the same time the spin of the electron was discovered. This remains the main change from pre-quantum physics, since the spin 1/2 has none classical equivalent. Pauli gave a wave equation for a non-relativistic equation with spin. This equation was the starting point of the attempt made by Dirac [19] to get a relativistic wave equation for the elec-tron. This Dirac equation was a very great success. Until now it is still considered as the wave equation for each particle with spin 1/2, electrons but also positrons, muons and anti-muons, neutrinos, quarks.

This wave equation was intensively studied by Louis de Broglie and his students. He published a first book in 1934 [15] explaining how this equation gives in the case of the hydrogen atom the quantification of energy levels, awaited quantum numbers, the true number of quantum states, the true energy levels and the Land factors. The main novelty in physics coming with the Dirac theory is the fact that the wave has not vector or tensor properties under a Lorentz rotation, the wave is a spinor and trans-forms very differently. It results from this transformation that the Dirac equation is form invariant under Lorentz rotations. This form invariance is the departure of our study and is the central thread of this work.

The Dirac equation was built from the Pauli equation and is based on 4 4 complex matrices, which were con-structed from the Pauli matrices. Many years after this first construction, D. Hestenes [20] used the Clifford al-gebra of space-time to get a different form of the same wave equation. Tensors which are constructed from the

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ENGLISH

equation for a magnetic monopole from this second gauge invariance [25]. He showed that a wave equation with a nonlinear mass term was possible for his magnetic monopole. When this mass term is null, the wave is made of two independent Weyl spinors.

This mass term is compatible with the electric gauge ruling the Dirac equation. So it can replace the linear term of the Dirac equation of the electron [3]. A nonlinear wave equation for the electron was awaited by de Bro-glie, because it was necessary to link the particle to the wave. But this does not explain how to choose the non-linearity. And the nonlinearity is a formidable problem in quantum physics: quantum theory is a linear theory, it is by solving the linear wave equation that the quantifica-tion of energy levels and quantum numbers are obtained in the hydrogen atom. If you start from a nonlinear wave equation, usually you will not be even able to get quanti-fication and quantum numbers.

Nevertheless the study of this nonlinear wave equa-tion began in the case where the Dirac equation is its linear approximation. In this case the wave equation is homogeneous. It is obtained from a Lagrangian density which differs from the Lagrangian of the linear theory only by the mass term. Therefore many results are simi-lar. For instance the dynamics of the electron are the same, and the electron follows the Lorentz force.

Two formalisms were available, the Dirac formal-ism with 4 4 complex matrices, and the real Clifford algebra of space-time. A matrix representation links these formalisms. Since the hydrogen case gave the main result, a first attempt was made to solve the non-linear equation in this case. Heinz Krger gave a pre-cious tool [23] by finding a way to separate the spherical coordinates. Moreover the beginning of this resolution by separation of variables was the same in the case of the linear Dirac equation and in the case of the nonlin-ear homogeneous equation. But then there was a great difficulty: The Yvon-Takabayasi angle is null in the x3 = 0 plane; This angle is a complicated function of an angular variable and of the radial variable; Moreover for any solution with a not constant radial polynomial, circles exist where the Yvon-Takabayasi angle is not defined; In the vicinity of these circles this angle is not small and the solutions of the Dirac equation have no reason to be linear approximations of solutions of the nonlinear homogeneous equation. Finally it was pos-sible to compute [4] another orthonormal set of solu-tions of the Dirac equation, which have everywhere a well defined and small Yvon-Takabayasi angle. These solutions are linear approximations of the solutions of the nonlinear equation. The existence of this set of or-thonormalized solutions is a powerful argument for our nonlinear wave equation.

When you have two formalisms for the same theory the question necessarily comes: which one is the best formalism? Comparing advantages of these formalisms, the possibility of a third formalism which could be the true one aroused. A third formalism is really available [6] to read the Dirac theory: it is the Clifford algebra of the physical space used by W.E. Baylis [1]. This Clifford al-gebra is isomorphic, as real algebra, to the matrix algebra generated by Pauli matrices. Quantum physics knew very early this formalism, since these Pauli matrices were in-vented to get the first wave equation with spin 1/2. Until now this formalism is also used to get the form invari-ance of the Dirac equation. Having then three formalisms for the same theory, the question was, once more: which is the true one?

The criterion of the best choice was necessarily the Lorentz invariance of the wave equation. Therefore a complete study, from the start, of this form invariance of the Dirac theory was made [8]. This problem was a classical one, treated by many books, but always with mathematical flaws. The reason is that two different Lie groups may have the same Lie algebra. The Lie algebra of a Lie group is the algebra generated by infinitesimal operators of the Lie group. Quantum mechanics uses only these infinitesimal operators and it is then very dif-ficult to avoid ambiguities. But it is possible to avoid any infinitesimal operator. And when you work without them you can easily see that the fundamental invariance group is larger than expected.

The first consequence of this larger group is the pos-sibility to define, from the Dirac wave, Lorentz dilations from an intrinsic space-time manifold to the usual rela-tivistic space-time. So the space-time is double and the Dirac wave is the link between these two manifolds. They are very different, the intrinsic manifold is not iso-tropic and has a torsion.

In several articles and in two previous books [11] [12], were presented several consequences of this larger invari-ance group. This invariance group governs not only the Dirac theory, but also all the electromagnetism, with or without magnetic monopoles, with or without photons. But it is not all of the thing, since this form invariance is also the rule for electro-weak and strong interactions [13].

Because it is impossible to read this paper without knowledge of the Cl3 algebra a first section presents Clif-ford algebras at an elementary level.

Section 2 reviews the Dirac equation, firstly with Di-rac matrices, where we get a mathematical correct form of the relativistic invariance of the theory. This neces-sitates the use of the space algebra Cl3. Next we explain the form of the Dirac equation in this simple frame, we review the relativistic form invariance of the Dirac wave. We explain with the tensors without derivative how the

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classical matrix formalism is deficient. We review plane waves. We present the invariant form of the wave equa-tion. Its scalar part is the Lagrangian density, another true novelty allowed by Clifford algebra. Finally we present the charge conjugation in this frame.

Section 3 introduces our homogeneous nonlinear equation and explains why this equation is better than the Dirac equation which is its linear approximation. We review its two gauge invariances. We explain why plane waves have only positive energy. The form of the spino-rial wave and the form of its relativistic invariance intro-duce the dilation generated by the wave from an intrinsic space-time manifold onto the usual relative space-time manifold, main geometric novelty of quantum physics. We explain the physical reason to normalize the wave. The link between the wave of the particle and the wave of the antiparticle coming from relativistic quantum me-chanics gives a charge conjugation where only the dif-ferential term of the wave equation changes sign. This makes the CPT theorem trivial and it is also a powerful argument for this wave equation. We get the quantifica-tion of the energy in the case of the hydrogen atom and all results of the linear theory with this homogeneous nonlinear wave equation.

Section 4 presents the invariance of electromagnetism under Cl3*, the group of the invertible elements in Cl3, for the Maxwell-de Broglie electromagnetism with massive photons, for the electromagnetism with magnetic mono-poles, for the four photons of de Broglie-Lochak.

1. Clifford algebrasThis section presents what is a Clifford algebra, then we study the algebra of an Euclidean plane and the algebra of the three-dimensional physical space which is also the algebra of the Pauli matrices. We put there the space-time and the relativistic invariance. Then we present the space-time algebra and the Dirac matrices. We finally present the Clifford algebra of a 6-dimensional space-time, needed by electro-weak and strong interactions.

It is quite usual in a physics book to put into appendi-ces mathematics even if they are necessary to understand the main part of the book. As it is impossible to expose the part containing physics without the Clifford algebras, we make here again a complete presentation of this nec-essary tool1.

We shall only speak here about Clifford algebras on the real field. Algebras on the complex field also exist and it could be expected to complex algebras to be key

1 Readers being in the know may do a quick review. On the con-trary a complete lecture is strongly advised for each reader who really wants to understand physics contained in the following sections.

point for quantum physics. The main algebra used here is also an algebra on the complex field, but it is its structure of real algebra which is useful in this frame2.

Our aim is not to say everything about any Clifford algebra but simply to give to our lecturer tools to under-stand the next sections.

1.1 What is a Clifford algebra?1 It is an algebra [2][11], there are two operations, not-ed A + B and AB, such as, for any A, B, C :

We shall only speak here about Clifford algebras on the real field. Algebrason the complex field also exist and it could be expected to complex algebras tobe key point for quantum physics. The main algebra used here is also an algebraon the complex field, but it is its structure of real algebra which is useful in thisframe.2

Our aim is not to say everything about any Clifford algebra but simply togive to our lecturer tools to understand the next sections.

1.1 What is a Clifford algebra?

1 - It is an algebra [2][11], there are two operations, noted A+B and AB, suchas, for any A, B, C :

(1.1)

A + (B + C) = (A + B) + C ; A + B = B + A

A + 0 = A ; A + (A) = 0A(B + C) = AB + AC ; (A + B)C = AC + BC

A(BC) = (AB)C.

2 - The algebra contains a set of vectors, noted with arrows, in which a scalarproduct exists and the intern Clifford multiplication uv is supposed to satisfyfor any vector u :

uu = u u. (1.2)

where u v is the usual notation for the scalar product of two vectors. 3 Thisimplies, since u u is a real number, that the algebra contains vectors but alsoreal numbers.

3 - Real numbers are commuting with any member of the algebra: if a is areal number and if A is any element in the algebra :

aA = Aa (1.3)

1A = A. (1.4)

Such an algebra exists for any finite-dimensional linear space which are theones that we need in this book.

The smaller one is single, to within an isomorphism.Remark 1: (1.1) and (1.4) imply that the algebra is itself a linear space, not

to be confused with the first one. If the initial linear space is n-dimensional, weget a Clifford algebra which is 2n-dimensional. We shall see for instance in 1.3that the Clifford algebra of the 3-dimensional physical space is a 8-dimensionallinear space on the real field. We do not need here to distinguish the left or

2A Clifford algebra on the real field has components of vectors which are real numbers andwhich cannot be multiplied by i. A Clifford algebra on the complex field has components ofvectors which are complex numbers and which can be multiplied by i.

3This equality seems strange, but gives nice properties. We need these properties in thenext sections.

5

(1.1)

2 The algebra contains a set of vectors, noted with arrows, in which a scalar product exists and the intern Clifford multiplication u v is supposed to satisfy for any vector u :





(1.1)

A + (B + C) = (A + B) + C ; A + B = B + A


A(BC) = (AB)C.


uu = u u. (1.2)



aA = Aa (1.3)

1A = A. (1.4)






5

(1.2)

where u v is the usual notation for the scalar product of two vectors3. This implies, since u u is a real number, that the algebra contains vectors but also real numbers.

3 Real numbers are commuting with any member of the algebra: if a is a real number and if A is any element in the algebra:





(1.1)

A + (B + C) = (A + B) + C ; A + B = B + A


A(BC) = (AB)C.


uu = u u. (1.2)



aA = Aa (1.3)

1A = A. (1.4)






5

(1.3)





(1.1)

A + (B + C) = (A + B) + C ; A + B = B + A


A(BC) = (AB)C.


uu = u u. (1.2)



aA = Aa (1.3)

1A = A. (1.4)






5

(1.4)

Such an algebra exists for any finite-dimensional lin-ear space which are the ones that we need in this book.

The smaller one is single, to within an isomorphism.Remark 1: (1.1) and (1.4) imply that the algebra is

itself a linear space, not to be confused with the first one. If the initial linear space is n-dimensional, we get a Clif-ford algebra which is 2n-dimensional. We shall see for instance in 1.3 that the Clifford algebra of the 3-dimen-sional physical space is a 8-dimensional linear space on the real field. We do not need here to distinguish the left or right linear space, since real numbers commute with each element of the algebra. We also do not need to con-

2 A Clifford algebra on the real field has components of vectors which are real numbers and which cannot be multiplied by i. A Clifford algebra on the complex field has components of vectors which are complex numbers and which can be multiplied by i.

3 This equality seems strange, but gives nice properties. We need these properties in the next sections.

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sider the multiplication by a real number as a third opera-tion, because it is a particular case of the multiplication.

Remark 2: If u and v are two orthogonal vectors, ( u v = 0), the equality ( u+ v)( u+ v) = ( u+ v)( u+ v) implies u u + u v + v u+ v v = u u+ u v + v u+ v v, so we get :

right linear space, since real numbers commute with each element of the algebra.We also do not need to consider the multiplication by a real number as a thirdoperation, because it is a particular case of the multiplication.

Remark 2: If u and v are two orthogonal vectors, (u v = 0), the equality(u+v) (u+v) = (u+v)(u+v) implies u u+u v+v u+v v = uu+uv+vu+vv,so we get :

0 = uv + vu ; vu = uv (1.5)Its the change to usual rules on numbers, the multiplication is not commuta-tive, we must be as careful as with matrix calculations.

Remark 3: The addition is defined in the whole algebra, which containsnumbers and vectors. So we shall get sums of numbers and vectors: 3 + 5i isauthorized. It is perhaps strange or disturbing, but we shall see next it is notdifferent from 3+5i. And everyone using complex numbers finally gets used to.

Even sub-algebra: Its the sub-algebra generated by the products ofan even number of vectors: uv, e1e2e3e4,...

Reversion: The reversion A A changes orders of products. Reversiondoes not change numbers a nor vectors a = a, u = u, and we get, for any u andv, A and B :

uv = vu ; AB = BA ; A+B = A+ B. (1.6)

1.2 Clifford algebra of an Euclidean plan: Cl2

Cl2 contains the real numbers and the vectors of an Euclidean plan, whichread u = xe1 + ye2, where e1 and e2 form an orthonormal basis of the plan:e1

2 = e22 = 1, e1 e2 = 0. Usually we set: e1e2 = i. The general element of

the algebra is :

A = a+ xe1 + ye2 + ib (1.7)

where a, x, y and b are real numbers. This is enough because :

e1i = e1(e1e2) = (e1e1)e2 = 1e2 = e2

e2i = e1 ; ie2 = e1 ; ie1 = e2i2 = ii = i(e1e2) = (ie1)e2 = e2e2 = 1 (1.8)

Two remarks must be made:1- The even sub-algebra Cl+2 is the set formed by all a + ib, so it is the

complex field. We may say that complex numbers are underlying as soon asthe dimension of the linear space is greater than one. This even sub-algebra iscommutative.

2 - The reversion is here the usual conjugation: i = e1e2 = e2e1 = iWe get then, for any u and any v in the plane: uv = u v+ i det(u,v) where

det(u,v) is the determinant.To establish that (u v)2 + [det(u,v)]2 = u 2v 2, it is possible to use uvvu

which can be calculated by two ways, and we can use vv which is a real numberand commutes with anything in the algebra.

6

. (1.5)

Its the change to usual rules on numbers, the multi-plication is not commutative, we must be as careful as with matrix calculations.

Remark 3: The addition is defined in the whole alge-bra, which contains numbers and vectors. So we shall get sums of numbers and vectors: 3 + 5 i

is authorized. It is

perhaps strange or disturbing, but we shall see next it is not different from 3 + 5i. And everyone using complex numbers finally gets used to.

Even sub-algebra: Its the sub-algebra generated by the products of an even number of vectors: u v, 1 2

3

4,...

Reversion: The reversion







uv = vu ; AB = BA ; A+B = A+ B. (1.6)




the algebra is :

A = a+ xe1 + ye2 + ib (1.7)


e1i = e1(e1e2) = (e1e1)e2 = 1e2 = e2







6

changes orders of products. Reversion does not change numbers a nor vec-tors a = a, u = u, and we get, for any u and v, A and B :







uv = vu ; AB = BA ; A+B = A+ B. (1.6)




the algebra is :

A = a+ xe1 + ye2 + ib (1.7)


e1i = e1(e1e2) = (e1e1)e2 = 1e2 = e2







6

(1.6)

1.2 Clifford algebra of an Euclidean plan: Cl2Cl2 contains the real numbers and the vectors of an Eu-clidean plan, which read u = x 1 + y

2, where

1 and

2

form an orthonormal basis of the plan:

12 = 22 = 1,

1 2 = 0.

Usually we set: 1

2 = i. The general element of the al-gebra is :







uv = vu ; AB = BA ; A+B = A+ B. (1.6)




the algebra is :

A = a+ xe1 + ye2 + ib (1.7)


e1i = e1(e1e2) = (e1e1)e2 = 1e2 = e2







6

(1.7)

where a, x, y and b are real numbers. This is enough be-cause :







uv = vu ; AB = BA ; A+B = A+ B. (1.6)




the algebra is :

A = a+ xe1 + ye2 + ib (1.7)


e1i = e1(e1e2) = (e1e1)e2 = 1e2 = e2







6

. (1.8)

Two remarks must be made:1 The even sub-algebra Cl2+ is the set formed by all

a + ib, so it is the complex field. We may say that com-plex numbers are underlying as soon as the dimension of the linear space is greater than one. This even sub-algebra is commutative.

2 The reversion is here the usual conjugation:







uv = vu ; AB = BA ; A+B = A+ B. (1.6)




the algebra is :

A = a+ xe1 + ye2 + ib (1.7)


e1i = e1(e1e2) = (e1e1)e2 = 1e2 = e2







6

We get then, for any u and any v in the plane: u v = u v + i det( u, v) where det( u, v) is the determinant.

To establish that ( u v)2 + [det( u, v)]2 = u2 v2, it is possible to use u v v u which can be calculated by two ways, and we can use v v which is a real number and commutes with anything in the algebra.

1.3 Clifford algebra of the physical space: Cl3Cl3 contains [1] the real numbers and the vectors of the physical space which read : u = x 1 + y

2 + z

3, where

x, y and z are real numbers, 1,

2 and

3 form an ortho-normal basis:

1.3 Clifford algebra of the physical space: Cl3

Cl3 contains [1] the real numbers and the vectors of the physical space whichread : u = xe1 + ye2 + ze3, where x, y and z are real numbers, e1, e2 and e3form an orthonormal basis:

e1 e2 = e2 e3 = e3 e1 = 0 ; e1 2 = e2 2 = e3 2 = 1. (1.9)

We let:

i1 = e2e3 ; i2 = e3e1 ; i3 = e1e2 ; i = e1e2e3. (1.10)

Then we get:

i21 = i22 = i

23 = i

2 = 1 (1.11)iu = ui ; iej = ij , j = 1, 2, 3. (1.12)

To satisfy (1.11) we can use the same way we used to get (1.8). To satisfy(1.12) we may firstly justify that i commutes with each ej .

The general element of Cl3 reads: A = a+u+iv+ib. This gives 1+3+3+1 =8 = 23 dimensions for Cl3.

Several remarks:1 - The center of Cl3 is the set of the a+ ib, only elements which commute

with every other ones in the algebra. It is isomorphic to the complex field.2 - The even sub-algebra Cl+3 is the set of the a + iv, isomorphic to the

quaternion field. Therefore quaternions are implicitly present into calculationsas soon as the dimension of the linear space is greater or equal to three. Thiseven sub-algebra is not commutative.

3 - A = a+uivib ; The reversion is the conjugation, for complex numbersbut also for the quaternions contained into Cl3.

4 - iv is what is usually called axial vector or pseudo-vector, whilst u isusually called vector. It is well known that it is specific to dimension 3.

5 - There are now four different terms with square -1, four ways to getcomplex numbers. Quantum theory is used to only one term with square -1.When complex numbers are used in quantum mechanics, it is necessary to askthe question of which i is used: i or i3 ?

1.3.1 Cross-product, orientation

uv is the cross-product of u and v. Using coordinates in the basis (e1, e2, e3),we can easily establish for any u and v:

uv = u v + i u v (1.13)(u v)2 + (u v)2 = u 2v 2 (1.14)

det(u,v, w) is the determinant whose columns contain the components ofvectors u, v, w, in the basis (e1, e2, e3). Again using coordinates, it is possible

7

1.3 Clifford algebra of the physical space: Cl3

Cl3 contains [1] the real numbers and the vectors of the physical space whichread : u = xe1 + ye2 + ze3, where x, y and z are real numbers, e1, e2 and e3form an orthonormal basis:

e1 e2 = e2 e3 = e3 e1 = 0 ; e1 2 = e2 2 = e3 2 = 1. (1.9)

We let:

i1 = e2e3 ; i2 = e3e1 ; i3 = e1e2 ; i = e1e2e3. (1.10)

Then we get:

i21 = i22 = i

23 = i

2 = 1 (1.11)iu = ui ; iej = ij , j = 1, 2, 3. (1.12)

To satisfy (1.11) we can use the same way we used to get (1.8). To satisfy(1.12) we may firstly justify that i commutes with each ej .

The general element of Cl3 reads: A = a+u+iv+ib. This gives 1+3+3+1 =8 = 23 dimensions for Cl3.

Several remarks:1 - The center of Cl3 is the set of the a+ ib, only elements which commute

with every other ones in the algebra. It is isomorphic to the complex field.2 - The even sub-algebra Cl+3 is the set of the a + iv, isomorphic to the

quaternion field. Therefore quaternions are implicitly present into calculationsas soon as the dimension of the linear space is greater or equal to three. Thiseven sub-algebra is not commutative.

3 - A = a+ui

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