1
Khmelnik S.I.
Wa e and pa icle – om dualism o uni y
Examples a e gi en when a mac oscopic objec mani es s i sel bo h as a wa e and as a angible
objec . I is p o en ha elemen a y pa icles a e bo h wa es and pa icles a he same ime, and no
al e na ely. This p oo is ob ained as new solu ions o Maxwell's equa ions. The p oo is no
comp ehensi e - only cubic, sphe ical and disk pa icles a e conside ed. This publica ion is a e iew and
addi ion o al eady published a icles and books.
Con en
1. In oduc ion
2. Cubic WAP
3. Sphe ical WAP
4. Disc WAP
5. Vacuum, da k ma e , da k ene gy
1. In oduc ion
In quan um physics, he e a e pos ula es ha ela e only o phenomena and objec s o he mic owo ld
and canno be applied in he wo ld o mac oobjec s.
And his app oach ga e b illian esul s (we will no lis hem).
We will ocus on only one (bu he main one o dis inguishing quan um physics om classical)
pos ula e, which decla es he exis ence o co puscula -wa e duali y, p ope ies o na u e, consis ing in he
ac ha ma e ial mic oscopic objec s can, unde some condi ions, exhibi he p ope ies o classical wa es,
and unde o he s, he p ope ies o classical pa icles. Jus in case, i was announced ha his p ope y is
also inhe en in la ge objec s, only i is in isible o hem.
Bu hen mac oscopic ball ligh ning was disco e ed [11, 12, 13, 14], which passes h ough he glass
like an elec omagne ic wa e, and a s a iona y and solid wa e o wa e appea s [20], on which ships a e
b oken. The e a e many hings in he wo ld ha sages ne e d eamed o am! We ha e o admi ha
co pucula -wa e dualism is a p ope y o all physical objec s. And he philosophical p inciple o duali y
will no help in unde s anding his p ope y, because we a e no longe in he magical wo ld o mic o-
objec s, whe e we can hope o he help o pos ula ed spells.
Any physical objec can mani es i sel bo h as a wa e and as a angible objec . And his ac equi es
explana ion (despi e he successes o quan um physics).
Below we will p o e ha pa icles a e bo h wa es and pa icles a he same ime, and no al e na ely.
The pa icle is a "wa e-AND-pa icle", no a "wa e-OR-pa icle". In wha ollows, we will use he
abb e ia ion WAP o wa e-AND-pa icle.
This p oo will be ob ained as new solu ions o Maxwell’s equa ions [1]. The p oo will no be
comp ehensi e: we will only conside he cubic pa icle, he sphe ical pa icle, and he disk pa icle. This
publica ion is a e iew and addi ion o al eady published a icles.
The e y idea ha such a model should exis is no new. E kin in [7] e iews a b ie his o y o his
idea. In 1900, he amous physicis and as onome J. Jeans a gued ha “in na u e he e a e wa es and
only wa es: closed wa es, which we call ma e , and open wa es, which we call adia ion o ligh ” [8].
E. Sch ödinge held he same iews un il he end o his li e, who w o e: “wha we now ake o pa icles
a e ac ually wa es” [9]. And he au ho o he concep o “wa e-pa icle” dualism, de B oglie, ini ially
also p oceeded om he ac ha “ he wa es desc ibed by quan um mechanics a e he sys em i sel ”
[10].
2. Cubic WAP [2].
2.1. Ma hema ical model o cubic WAP
Conside some olume V wi h magne ic pe meabili y and dielec ic cons an . Le , as a esul o
some in luence, an elec omagne ic wa e wi h ene gy a ise in his olume. The e is no hea loss in
olume V and he e is no adia ion om i . A e some ime, he wa e pa ame e s will ake on s a iona y
2
alues, de e mined by he alues , , and olume size. These pa ame e s a e he elec ic ield s eng h
and he magne ic ield s eng h as a unc ion o Ca esian coo dina es and ime,
i.e. (,,,) and (,,,). Na u ally, hey sa is y he sys em o Maxwell equa ions o he o m
−
−
=0, (1)
−
−
=0, (2)
−
−
=0, (3)
−
+
=0, (4)
−
+
=0, (5)
−
+
=0, (6)
+
+
=0, (7)
+
+
=0. (8)
Conside he ollowing unc ions (p oposed in [3]) ha sa is y his sys em o equa ions:
(,,,)=cos()sin()sin()sin(), (9)
(,,,)=sin()cos()sin()sin(), (10)
(,,,)=sin()sin()cos()sin(), (11)
(,,,)=ℎsin()cos()cos()cos(), (12)
(,,,)=ℎcos()sin()cos()cos(), (13)
(,,,)=ℎcos()cos()sin()cos(), (14)
whe e,,,ℎ,ℎ,ℎ- cons an ampli udes o unc ions, - cons an s. Di e en ia ing (9-14) and
subs i u ing he esul in o (1-8), a e educing he common ac o s, we ob ain:, , ,
ℎ−ℎ+=0, (15)
ℎ−ℎ+=0, (16)
ℎ−ℎ+=0, (17)
−−ℎ=0, (18)
−−ℎ=0, (19)
−−ℎ=0, (20)
++=0, (21)
ℎ+ℎ+ℎ=0. (22)
Le us conside he solu ion o he esul ing sys em o equa ions ound in [4]. Since he sys em is
symme ic, we accep ==. (23)
In his case, he sys em o equa ions (15-22) akes he o m:
ℎ−ℎ+
⁄=0, (24)
ℎ−ℎ+
⁄=0, (25)
ℎ−ℎ+
⁄=0, (26)
−−ℎ
⁄=0, (27)
−−ℎ
⁄=0, (28)
−−ℎ
⁄=0, (29)
++=0, (30)
ℎ+ℎ+ℎ=0. (31)
In he sys em o equa ions (24-31), equa ions (30, 31) ollow di ec ly om he p e ious ones. Indeed,
adding equa ions (27-29), we ge (31), and adding (24-26), we ge (30). The i s 6 equa ions in he sys em
(24-31) wi h 6 unknowns a e independen and om hem he ampli udes o he unc ions
,,,ℎ,ℎ,ℎ can be ound. We will look o a solu ion o sys em (24-29) a
ℎ=0. (32)
In his case we ind:
3
ℎ=−ℎ, (33)
=−
, (34)
=, (35)
=−2, (36)
=−
. (37)
F om (34, 37) we ind: =
. (38)
F om (34, 38) we ind: =−
=−ℎ
, (39)
o ℎ=−
. (40)
2.2. Ene gy WAP
Le us w i e he ensions (9-14) in he o m
=(,,,)
(,,,)
(,,,)=
cos()sin()sin()
sin()cos()sin()
sin()sin()cos()sin(), (41)
=(,,,)
(,,,)
(,,,)=ℎ
ℎ
ℎsin()cos()cos()
cos()sin()cos()
cos()cos()sin()cos(). (42)
Le us deno e he ime-independen pa s o hese exp essions:
=(,,)
(,,)
(,,)=
cos()sin()sin()
sin()cos()sin()
sin()sin()cos(), (43)
=(,,)
(,,)
(,,)=ℎ
ℎ
ℎsin()cos()cos()
cos()sin()cos()
cos()cos()sin(). (44)
Le us now ind he squa ed modulus o he o al ensions:
=++, (45)
=++. (46)
F om (45-46) we ind: =++sin(), (49)
=++cos(). (50)
Le 's deno e: ||=++, (51)
||=++/ (52)
Then we ge : =(||sin()), (53)
=(||cos()). (54)
Le us ind and . Fi s o all, we will show ha he e exis s a pa allelepiped in which he
o al ene gy emains cons an in ime. Le he segmen s OA and OB on he oz axis ha e equal leng h Z,
which mee s he condi ion. ∙
=,. (55)
whe e is in ege . Ob iously, he condi ion is sa is ied
4
∫cos()
=∫sin()=.
(56)
Le us conside a olume in which condi ions simila o (55, 56) a e sa is ied along any coo dina e,
and we will call such a olume a ag eed olume. Le 's ind he alue o he ag eed olume. F om (55) we
ind he leng h om he coo dina es:
2=2
⁄,2=2
⁄ ,2=2
⁄. (57)
Then he o al ag eed olume =8=8
⁄, (58)
and he minimum ag eed olume
=8
⁄ (59)
o , aking in o accoun (38), =8
.
. (60)
y x
o
B
A
z
Fig. 1.
Le us w i e exp essions (43, 44) using he solu ion ob ained abo e (32, 33, 35, 40):
=(,,)
(,,)
(,,)=11
−2, (61)
=(,,)
(,,)
(,,)=
−1
10, (62)
whe e =(,,)
(,,)
(,,)=cos()sin()sin()
sin()cos()sin()
sin()sin()cos(), (63)
=(,,)
(,,)
(,,)=sin()cos()cos()
cos()sin()cos()
cos()cos()sin() (64)
F om (51, 61, 63) we ob ain:
||=++=11
−2=114=cos()sin()sin()
sin()cos()sin()
sin()sin()()114=
5
⎩
⎨
⎧
cos()sin()sin()+
sin()cos()sin()+
4sin()sin()()
⎭
⎬
⎫
o ||=6(). (65)
The las ans o ma ion ollows om (56). Simila ly, om (52, 62, 64, 56) we ob ain:
||=++=
−1
10=
110=sin()cos()cos()
cos()sin()cos()
cos()cos()sin()
110=
sin()cos()cos()+
cos()sin()cos()+
0 (66)
o ||=
2(). (66)
Thus, o he ag eed olume om (65, 66) we ob ain:
||||=
. (67)
F om (65-67) i ollows: =||=||=6(). (68)
The ene gy densi y is =+. (69)
F om (53, 54, 69) we ge : =||sin()+||cos(). (70)
F om (68, 70) i ollows ha =(sin()+cos())=, (71)
i.e. in a ag eed olume, he ene gy densi y in he olume does no depend on ime and has a cons an alue
h oughou he en i e WAP olume. In o he wo ds, a s anding wa e is c ea ed in a ag eed olume ha
does no adia e. The alue is a cons an . The e o e, o a ag eed olume, he exp ession o he ene gy
in he en i e olume is =∙. (72)
Fo a minimum olume o WAP, as ollows om (68),
==6. (72a)
F om (72, 72a, 60) we ind he ene gy o he minimum olume o WAP:
=6∙8
.
=∙
⁄, (73)
whe e =483...=2.4∙10... (74)
Consequen ly, in a cons an ag eed olume, he ene gy o an elec omagne ic wa e does no depend
on ime, i.e. emains cons an . This means ha unde he speci ied condi ions,
S a emen 1. WAP, like a s anding elec omagne ic wa e, can exis in a ag eed olume.
2.3. Flows o ene gy
Ene gy lux densi ies along coo dina es a e de e mined by he o mula
=
=(×)=−
−
−, (75)
whe e he unc ions , a e de e mined om (9-14). Ob iously, in a consis en olume a he bounda ies
o he coo dina e axes he ollowing condi ions a e sa is ied:
sin()=sin()=sin(). (76)
6
The unc ion is p esen in he de ini ion o one o he unc ions speci ied in condi ion (75). The e o e, om
(75, 76) i ollows ha ene gy lows di ec ed pe pendicula o he aces a e equal o ze o, i.e. his olume
does no exchange ene gy wi h he en i onmen .sin
S a emen 2. WAP can exis wi hin an ag eed olume.
In addi ion, o such a olume, S a emen 1 is sa is ied. Thus, WAP can exis in such a olume. Fi s
o all, le us conside he cubic o m p oposed in [4]. Conside , o example, he ene gy lux densi y along
he axis . F om (75) we ind: =− (77)
Combining his o mula wi h o mulas (9, 10, 12, 13, 23), we ind:
=(sin()cos()cos()ℎsin()cos()sin()
−cos()sin()cos()ℎcos()sin()sin())sin(2)
Taking in o accoun (33, 35, 40), om (77) we ob ain:
=sin()cos()cos()3/sin()cos()sin()
−cos()sin()cos()3/cos()sin()sin()sin(2)
o =3
sin()cos()cos()sin()cos()sin()
+cos()sin()cos()cos()sin()sin()sin(2)
o =
sin(2)sin()cos()
+cos()sin()sin(2) (78)
o =
sin(2)sin(2) (78)
o =
sin(4+4), (79)
We ha e ob ained an equa ion o he ene gy lux densi y along he axis . This lux a ies wi h ime. I is
equal o ze o on he aces o he cube in he case when on he aces o he cube, i.e. when = (see Fig.
1) condi ions o he o m sin(2)=0 a e me . These condi ions a e me o he ag eed ex en - see (55).
Le us conside he ene gy lux densi y along he axis . F om (75) we ind:
=− (80)
Combining his o mula wi h o mulas (10, 11, 13, 23), we ind:
=12−sin()sin()cos()ℎcos()sin()cos()sin(2)
Taking in o accoun (35, 32, 36, 33, 40), om (80) we ob ain:
=12−2sin()sin()cos()3/cos()sin()cos()sin(2)
o =−
sin(2)sin()sin(2)sin(2) (81)
Since on he aces o he cube sin(2)=0, hen on he aces o he cube =0. Conside he ene gy
lux densi y along he axis . F om (75) we ind:
=− (82)
Combining his o mula wi h o mulas (9, 11, 12, 23), we ind:
=sin()sin()cos()ℎsin()cos()cos()sin(2)
Taking in o accoun (36, 33, 40), om (82) we ob ain:
7
=2sin()sin()cos()3
sin()cos()cos()sin(2)
o =
sin()sin(2)cos()sin(2) (83)
F om equa ions (78, 81, 83) i ollows ha lows o elec omagne ic ene gy ci cula e in he cube along all
axes. species (78, 84, 85).
Conside he ec o sum =
+
+
. (84)
Ob iously, many ec o s ci cula e in he cube and a each poin o he cube he e is a ce ain ec o
ha has a module - he densi y o he o al ec o o he elec omagne ic ene gy low. F om (78, 81,
83, 75, 63, 64) i ollows ha
=
=−
−
−sin(2). (85)
F om (78, 81, 83, 85) i ollows ha
=о
sin(2), (86)
whe e о
=−+−+−. (87)
Thus, inside he cube he e a e lines o med by ec o s . Ob iously, such a line ep esen s some
kind o “spa ial en angled spi al” (he eina e simply a spi al). Such spi als a e closed. Th ough e e y
poin whe e о
≠0 he e is a single spi al, and h ough e e y poin whe e о
≠0 he e a e many spi als.
A each poin o his spi al, he magni ude o he low luc ua es in ime, as sin(2). The ampli ude
o hese luc ua ions changes a a gi en poin and depends on he loca ion o his poin in he cube.
You can conside he de elopmen o his spi al. Le us deno e he coo dina e o a poin on his scan
as . Then we ge a sinusoid wi h an ampli ude ha is a unc ion o his coo dina e:
(,)=о()∙sin(2), (88)
whe e ,о is a mo e con enien no a ion o unc ions ,о
, espec i ely.
Le 's expand he unc ion о() in o a igonome ic se ies:
о()=оо+∑ (оsin())
(89)
Acco dingly, unc ion (88) will ake he o m:
=ооsin(2)+∑ (оsin()sin(2))
. (90)
Each e m o his sum can be ep esen ed as:
оsin()sin(2)=оsin()cos 2−=+, (91)
Whe e =оsin−+2, (92)
=оsin+−2. (93)
Each o hese wo ea u es a a eling wa e. Consequen ly, he unc ion unde conside a ion (90)
ep esen s he sum o many a eling wa es o elec omagne ic ene gy low. So, many unning wa es o
ene gy low ci cula e along each spi al. These wa es ha e a common equency, bu di e in di ec ion o
mo emen , phase and ampli ude. The o al ampli ude o he low o hese wa es is equal o
спираль= ∑о
(94)
8
2.4. Weigh
In he exis ing heo y, elec omagne ic mass is he mass o an elec omagne ic wa e ha is c ea ed
by a mo ing pa icle [5]. In ou case, i is he wa e ha c ea es he WAP pa icle, and in his wa e he e
a e no pa icles ha o m i . Bu a he same ime, we canno use his app oach o de e mine he mass.
We will use he well-known Umo o mula, which connec s he ene gy densi y and ene gy low
wi h he speed o ene gy mo emen :
=
. (95)
I is also known ha he pulse densi y
=
, (96)
and he mass ==
. (97)
Hence, =
. (98)
In his case, o a wa e wi h known in ensi ies, one can ind he ene gy densi y , elec omagne ic ene gy
lux densi y and mass densi y acco ding o (98).
I is shown abo e ha in he cube he e a e ajec o ies along which lows o elec omagne ic ene gy
p opaga e. A he same ime, many such lows pass h ough each poin o he WAP cube. Le us deno e
he o al powe densi y o such lows as . Then, using (98), we ind he densi y o he elec omagne ic
mass, which is gene a ed a his poin by he e y exis ence o he elec omagne ic wa e in he WAP. The
sum o hese masses is he elec omagne ic mass o WAP.
Consequen ly, WAP can be conside ed bo h as a s anding wa e and as a olume ha ing a ce ain
mass.
2.5. Conclusion
We ha e es ablished wo condi ions ha mus be sa is ied by he egion in which
WAP can exis wi hin a closed and con inuous bounda y.
WAP, like a s anding elec omagne ic wa e, can exis in a consis en olume
We ha e es ablished ha WAP o ms a closed a ea and has a ce ain shape and olume. The esul s
ob ained can be applied o any a bi a ily small uni s o leng h. The shape o he WAP egion is such ha
mul iple WAPs can be adjacen o each o he wi hou gaps. Consequen ly, WAP g oups can occupy any
olume. Thus, WAP o any size and a eas o WAP o any size can exis . WAP does no ha e i s own speed
and i s mechanical ene gy is de e mined by i s mass and he speed ha i ecei ed when in e ac ing wi h
o he masses (including o he WAP). The in e nal p essu e on he WAP bo de is equal o he ene gy
densi y a he bo de , al hough WAP does no ha e any en elope. I can be assumed ha WAP beha es
like an absolu ely elas ic body and ansmi s he ecei ed impulse wi hou changing i s magni ude. Then
he WAP egion also beha es as a conduc o o he impulse. Ob iously, WAP can o m elemen a y
pa icles and la ge s uc u es. Bu we can assume ha he acuum is also wo en om WAP.
3. Sphe ical WAP[6].
3.1. Maxwell's equa ions in sphe ical coo dina es
In [1], a solu ion o Maxwell's equa ions in sphe ical coo dina es was ound. The known solu ion o
a sphe ical elec omagne ic wa e does no sa is y he law o conse a ion o ene gy (i is conse ed only
on a e age), he elec ic and magne ic in ensi ies o he same name (in coo dina es) a e in phase, only one
o Maxwell’s sys em o equa ions is sa is ied, he solu ion is no a wa e one, he e is no ene gy low wi h
a eal alue. The p oposed solu ion is ee om hese sho comings. Maxwell's sys em o equa ions, being
a sys em o pa ial di e en ial equa ions, has many solu ions. The applicabili y o a solu ion o physics is
de e mined by a single c i e ion: i mus sa is y he law o conse a ion o ene gy (LEC). The exis ing
solu ion does NOT sa is y his law.
So, le 's conside he sys em o Maxwell's equa ions o acuum, which has he o m
o ()+
=0, (1)
9
o ()−
=0, (2)
di ()=0, (3)
di ()=0. (4)
whe e is he elec ic ield s eng h, is he magne ic ield s eng h, is he absolu e magne ic
pe meabili y, is he absolu e dielec ic cons an . Nex , sphe ical coo dina es a e conside ed - see Fig. 1.
Maxwell's equa ions in sphe ical coo dina es in he absence o cha ges and cu en s ha e he o m gi en
in able. 1.
Fig. 1.
Table 1.
1
3
1
g
(
)
+
−
sin
(
)
+
=
0
2
sin
(
)
−
−
+
=
0
3
+
−
+
=
0
4
+
+
g
(
)
+
+
sin
(
)
=
0
5
(
)
+
−
(
)
−
=
0
6
(
)
−
−
−
=
0
7
+
−
−
=
0
8
+
+
(
)
+
+
(
)
=
0
In he solu ions ound, he ensions a e de e mined by o mulas o he ollowing o m:
=
Khm(,∝)sin(∝++), (5)
=
Khm(,∝)cos(∝++ ) , (6)
=
Khm(,∝)sin(∝++ ) , (7)
=
Khm(,∝)cos(∝++), (8)
=
Khm(,∝)sin(∝++), (9)
=
Khm(,∝)cos(∝+ + ) , (10)
whe e Khm is some unc ion, ∝,,,,ℎ a e cons an s. We will conside a special case when
16
We will no conside he exis ing explana ions o his phenomenon. Bu his expe imen is amazing: a
b igh sphe ical adian cold a ea, he appea ance o which is inexplicable.
A pulse wi h a s eep leading edge can be expanded in a Fou ie se ies, whe e a sinusoidal unc ion
wi h a high equency will p e ail. Thus, i can be assumed ha he capaci o is connec ed o a high- ol age
and high- equency gene a o . No elemen s a e connec ed in se ies wi h he capaci o , i.e. i is loca ed
absolu ely symme ically ela i e o he gene a o e minals. In his case, he ene gy low in o he capaci o
comes om wo sides. Two equal and opposi ely di ec ed ene gy lows mee exac ly a he cen e o he
capaci o . Abo e we conside ed a neu ino ha was o med when wo iden ical wa es me . The objec we
obse e in his expe imen can be called a "gian neu ino."
Fig. 4.
I is p oposed o conside neu inos as DWAP. I ully co esponds o he abo e desc ip ion o
neu inos. Abo e we de ined o i (mo e p ecisely, o he cylinde -disk, which makes up hal o he
neu ino)
ene gy
acco ding o (13),
mass =
с – see (9),
angula momen um =
с acco ding o (17).
4.6. Mo e abou neu inos
Abo e, we examined a neu ino, which was o med when wo iden ical wa es me , in which all
cha ac e is ics coincided, excep o he di ec ion o ligh and di ec ion o o a ion. This is, o cou se, an
unlikely case. Now conside he gene al case when he pa ame e s ,, di e and deno e hem o he
i s and second wa es as ,, and ,,. In his case, om (12, 13) we ob ain:
W
()=
, (21)
()=
с
, (22)
W
()=
, (23)
()=
с
. (24)
A e he wa es mee , he newly o med pa icle lies owa ds a mo e massi e wa e (wi h he same speed
с), and bo h hal es o i ( o a ing, as be o e, in opposi e di ec ions wi h he same speeds) acqui e a new
pa ame e alue and a new alue o he angula eloci y o o a ion
ha is common o bo h hal es .
We assume ha he newly o med neu on lies owa ds he i s wa e. In his case, he kine ic powe o
he second hal W
=0. Acco ding o he law o conse a ion o ene gy, simila ly o (3.1), we ind:
W
()+
()+ W
()+
()=
W
()+
()+
() , (25)
whe e
17
W
()=
, (26)
()=
с
, (27)
()=
с
. (28)
Fo mula (25) is an equa ion wi h one unknown . In his case, he o al ene gy o he pai emains
cons an . F om he e and om (9) i ollows ha he o al mass o he pai also emains cons an , i.e. he
appea ance o a neu on does no change he a io o mass and ene gy.
5. Vacuum, da k ma e , da k ene gy [21].
5.1. In oduc ion
The s uc u e o he acuum is s udied by quan um ield heo y, which ne e ge s i ed make i look
e y complica ed and, indeed, does no o e any hing o desc ibe he s uc u e o he acuum ha is
consis en wi h he ideas o classical physics. Below we p opose such a s uc u e, which ollows only om
he solu ion o Maxwell’s equa ions - no addi ional assump ions a e made. This s uc u e can be he
s uc u e o acuum, da k ma e , da k ene gy, any egion o space... He e we will no es ablish he scope
o applica ion o his s uc u e. On he con a y, he au ho would like o hea a discussion o his idea,
which was ou lined back in 2020 [22] (Chap e 5). Bu he public is s e nly silen .
I was p o en abo e ha he e can be a cubic WAP, which is a cubic olume o acuum, and in
which a s anding olume ic wa e pulsa es. I is impo an o no e ha his olume does NOT ha e any
bounda ies - physical o o med by he he e ogenei y o he en i onmen . WAP does NOT adia e h ough
he aces o he cube, bu on each ace he e is an elec ical in ensi y, he ec o o which is di ec ed
pe pendicula o his ace. The amoun o ene gy, equency, and ension a he cube's aces a e unc ions
o he size o he cube only. Appa en ly, he e is a smalles olume o a cube, de e mined by he minimum
ene gy quan um.
Many o hese WAP can ill he space en i ely, wi hou gaps. And i is p ecisely his s uc u e ha is
desc ibed below. This s uc u e occu s in na u e [20] – in Fig. 1 and Fig. 2 show so-called squa e wa es
on he sea.
Fig. 1.
Fig. 2.
18
In [21, 22], Chap e 5, i is shown ha he e a e se e al a ian s o squa e WAP. In Fig. Figu e 1
shows one o he op ions - magne ic s eng h eme ging om he aces o he cube a e shown. I is
impo an o no e ha he e is no ension in his case, al hough i is shown in Fig. 3. On aces wi h a
nega i e alue, he s ess coo dina es a e di ec ed in he nega i e di ec ion.
The ene gy low does no lea e he ace pe pendicula o he x axis, bu ci cula es along his ace,
because lux densi ies and on his ace a e no equal o ze o. Fo example, =−.
He e ≠0. Consequen ly, on his ace, as well as in he en i e olume, he e is ene gy wi h a densi y
ha does no change o e ime. Consequen ly, on his ace and, in gene al, on all aces, he e is a cons an
p essu e equal o he ene gy densi y.
х
у
Нх
Нх
L=pi/al a
Нy
Нy
Нz
Нz
Fig. 3.
5.2. Vacuum s uc u e
Le us now conside he se o WAP. The cubic shape o WAP sugges s ha many WAP o m a
con inuous olume - see ig. 4. Va ious combina ions o WAP a e possible.
The e may be a space illed wi h WAP, c ea ing only magne ic s eng hs on he edges o only
elec ical s eng hs on he edges.
The e may be a space illed wi h only symme ical WAPs o only asymme ical WAPs. In he la e
case, a di ec ion should a ise in space in which he e is no ension in any di ec ion. Such a acuum mus
somehow exhibi aniso opic p ope ies.
I can be assumed ha na u e uses all op ions and he e a e he e ogeneous spaces.
Thus, each WAP emains au onomous, bu oge he hey o m a con inuous olume o acuum.
I can be assumed ha all WAP ha e he same olume and hen he e is a single acuum equency.
I can also be assumed ha he e a e di e en egions o space wi h di e en (bu common o a gi en
egion) olume o WAP. Then hese a eas should ha e di e en acuum equencies.
Fig. 4.
19
Any ace o WAP may end up on he bo de o an emp y egion o space. Then ension will a ise
on he bo de o his a ea - he ension ha is p esen on he speci ied bo de o WAP. This ension is he
gi en ension ha o ms a s anding wa e. Thus, he ension on he edge o some WAP gene a es a s anding
wa e in he emp y space and he eby c ea es a new WAP. In his way, WAP mul iplies, illing he en i e
acuum. I can be assumed ha he Uni e se a ose om one WAP.
5.3. Casimi e ec
Le us conside he igh side su ace o he acuum agmen in Fig. 3. Assume ha his su ace is
he bounda y o he WAP egion. On he open su aces o WAP in hei cen e he ec o s o ensions
en e ing and lea ing hese su aces a e shown. The hick line going a ound he ends o hese ec o s
con en ionally depic s a wa e o ension on open su aces. These ensions a y sinusoidally in ime. Thus,
he e is a s anding wa e o ensions on he su ace o he WAP domain bo de .
Bu , mos impo an ly, he e is cons an p essu e on exposed WAP su aces. I some body is adjacen
o hese su aces, hen i mus expe ience his p essu e. Thus, he body, loca ed in a acuum illed wi h
WAP, expe iences acuum p essu e om all sides. Each WAP a ea also pu s p essu e on he neighbo ing
a ea. Consequen ly, WAP seeks o ill in e nal oids. One can a gue, ollowing To icelli, ha “a acuum
does no ole a e a emp iness.”
In mo e de ail, he book p o es ha wha has been said is no hing mo e han a p oposed explana ion
o he Casimi e ec - wo pa allel mi o su aces loca ed a sho dis ances in a acuum a ac each o he .
In he exis ing acuum model, he cause o he Casimi e ec is conside ed o be “ene gy
luc ua ionsphysical acuumdue o cons an bi h and disappea ance in i i ual pa icles…. This occu s
due o he ac ha only s anding wa es can exis in he space be ween he pla es, he ampli ude o which
on he pla es is ze o. As a esul , he p essu e o i ual pho ons om he inside on he wo su aces u ns
ou o be less han he p essu e on hem om he ou side, whe e he bi h o pho ons is no limi ed in any
way.“In addi ion, when explaining his e ec , he exis ence o nega i e ene gy is ecognized. These
e e ences a e p o ided o highligh he appa en con adic ion be ween he p oposed and exis ing heo ies
(PT and ET).
In PT i is p o ed ha he e is a olume ic s anding wa e wi h ce ain in ensi ies a he nodes, and
in ST i is s a ed ha he ampli ude o he in ensi ies a he nodes (on he pla es) is equal o ze o (i can be
p o en ha he law o conse a ion o ene gy is no sa is ied in his case).
In PT i is p o ed ha eal pa icles ill he acuum, and in ET he exis ence o i ual pa icles is
assumed, he bi h o which is no limi ed by any hing, and he disappea ance o which is inexplicable.
In PT i is p o en ha he e is a cons an acuum p essu e on bodies, and in ET i is assumed ha
such p essu e is c ea ed by wa es o i ual pa icles ha cons an ly appea and disappea .
The ET p o es he exis ence o nega i e ene gy, while he PT main ains espec o he law o
conse a ion o ene gy.
The eade is in i ed o choose wha he likes bes .
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