The Quantum Mechanical Time Reversal Operator

8/13/2019 The Quantum Mechanical Time Reversal Operator

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The Quantum Mechanical Time Reversal Operator.

1. Introduction.

Callender [1] argues for two contentious conclusions, both of which I support: that

non-relativistic quantum mechanics is irreversible (non-time reversal invariant, or

non-!I for short", both in its probabilistic laws, and in its deterministic laws#

hese claims contradict the current assumptions in the sub$ect# he first point, the

irreversibilit% of the probabilistic part of quantum mechanics, is the most

important for understanding irreversible processes, and was alread% argued

convincingl% some fift% %ears ago b% &atanabe in 1' [)], [*], as confirmed b%

+eale% [] and +olster [], although it has been overlooed b% most authorities on

the sub$ect, such as .avies [/], 0achs [], and 2eh [3]# 0imilar points have also

been made independentl%, notabl% b% 0chrodinger ['] and 4enrose [15]# 6ut I will

not deal with this problem here, since I have e7amined it in detail elsewhere []#

+ere I onl% e7amine Callender8s second claim, i#e# that the deterministic

d%namics of quantum mechanics is also non-time reversal invariant# he problem

here is more subtle, and we will find that the orthodo7 anal%sis suffers from a

number of deep-seated conceptual confusions#

Callender [1], distinguishes between TRI (ime !eversal Invariance" and

WRI (&igner !eversal Invariance", the latter being generall% interpretedas time

reversal invariance in quantum mechanics, while the former is generall% dismissed

in quantum mechanics as logicall% incoherent# TRI is s%mmetr% under the simple

transformation: T: t-t alone# WRIis s%mmetr% under the combined operation:

= T*, where T is the simple time reversal operator, and 9 is comple7

con$ugation# Callender first observes that:

If one surve%s the literature concerning this issue, one finds man% arguments that attempt to

blur the difference between &!I and !I# 4robabl% the most frequent claim is that in

quantum mechanics the ph%sical content is e7hausted b% the probabilities# ;s .avies puts it,


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;fter briefl% dismissing this ind of argument, Callender then observes:

his is not the place to go through all the misguided attempts to blur the distinction between

&!I and !I, but another popular argument claims that &!I is necessitated b% the need to

switch sign of momentum and spin under time reversal# +ere the repl% is that there is no such

necessitation# In quantum mechanics, momentum is a spatial derivative (-i @7) and spin

is a ind of


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theor%" T exactlyis the geometrical reversal of the time a7is which is appropriate in the four

dimensional space-time geometr%, and is thus naturally akin to the Feynann !i"!a"

philosophy.[11], p#)#

I will follow de 6eauregard and refer to the transformation: T: t-tapplied to

quantum states as theRacah operator, and the orthodo7 T* as the Wi"ner

operator. de 6eauregard8s anal%sis is e7tremel% interesting, but it is about

relativistic quantum mechanics, and he does not argue that T is appropriate in non-

relativistic quantum mechanics, or anal%se the underl%ing logic of this choice in

an% detail, which is the aim here# I comment on his views in the final section#

I will define the deterministic part of ordinar%, non-relativistic quantum

mechanics as 8, and the first point is that:

Wigner Invariance: T*(=>" G =>, or equivalentl%: T(=>" G 9(=>"

Racah Non-Invariance: T(=>" =>, and: *(=>" =>

It is readil% seen that the time dependant 0chrodinger equation is unchanged b%

the transformation T*,but changed to an anti-s%mmetric form b% T alone, and b%9 alone, b% looing at the simple 0chrodinger equation for a free particle, and its

transformations:

Theory: Images of Schrodinger Equation: Simple Solutions

=> @t G i @)m )@7) ; e7p((i@ "(px-p#t$)m""

T(=>" -@t G i @)m )@7) ; e7p((i@ "(px%p#t$)m""

T9(=>" -@t G -i

@)m )

@7)

; e7p((i@

"(-px-p#

t$#""

9(=>" @t G -i @)m )@7) ; e7p((i@ "(-px%p#t$#m""

he ore realisticall%, free particles are


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of this paper# he class of these simple solutions for T*(=>" is the same as for

=> becausep can be positive or negative# 6ut the class of solutions for *(=>"

(or equall% T(=>"" is not the same as for => becausep)must be positive#

o see the main relationships between the transformations, we can tae =>

(or equivalentl%, T*(=>" " to represent a class J of solutions to the

0chrodinger wave equation, and T(=>" (or equivalentl%, 9(=>" " to represent a

", which is identical to the set of solutions to 9(=>"# his is dis$oint

from =>#

he wave functions in Eig# 1 have also been stratified into three inds:

&'ll possible (ave unctions.)

=> G T9(=>"

; 6 C

! !T9

T! !9

C8 68 ;8

T(=>" G 9(=>"


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; represents non-equilibrium thermod%namic processes (or retarded

waves, dispersing from a centraliBed source"#

6 represents equilibrium thermod%namic processes (with ma7imum

dispersion"#

C represents non-equilibrium with waves from 9(=>", when we combine different particles into composite

s%stems (or we would get the wrong inds of interference effects"# 6ut the choice

to use the class=> rather than the class9(=>" (or equivalentl% (=>"" torepresent particles can be regarded as an arbitrar% convention in the first place# If

0chrodinger had chosen to use 9(=>" instead of =>, then we would simpl% have

to "

can be used to represent a perfectl% sensible theor%, isomorphic to =>#

Aow let us e7amine the deterministic laws satisfied b% 9(=>", or

equivalentl% T(=>", rather than =># he transformed wave equation is as given


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above: it is anti-s%mmetric with the usual 0chrodinger equation for =># Eor the

free particle in =>:

(1" @t G i

@)m

)

@7

)

In the T-transformed theor%, we have instead:

T(1" -@t G i @)m )@7)

his is obvious and well recogniBed# 6ut what of the laws for the energ% and

momentum operatorsN hese relate the classical properties of energ% andmomentum to the wave functions# In ordinar% => the e% laws are:

()" "G i @t Oinetic Pnerg% (Bero potential"

(*" #G -i @7 >omentum

;long with:

(" "G #)@)m Classical Pnerg%->omentum !elation

;nd then Pqs# * and entail:

(" "G - )@)m )@7)

0ubstituted in Pq# ) this gives: - )@)m )@7)G i @t, which is $ust Pq# 1

rearranged#

hen what are the tie reversed ia"esof Pqs#)-N Eor the ", or equivalentl% T(=>", the operators should be given b%:

T()" "$G -i @t Oinetic Pnerg% (with Bero potential"

T(*" #$G i @7 >omentum

T(" "$G #$)@)m Classical Pnerg%->omentum !elation

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I have labeled these "$and #$, to mae clear that these are distinct mathematical

operators to " and #M the% are what these operators defined as giving the classical

energ% and momentum transform to in the reversed theor%#

&e will wor through this in more detail later, but it is eas% enough to see wh%

these must be adopted# In 9(=>", we tae the wave: 9 to represent a particle with

the same classical properties as in => M this is the basic isomorphism#

;lternativel%, in T(=>", Trepresents a particle with the time-reversed classical

properties represented b% in =># Aow, for instance, consider the special solution,

,stated earlier for =># Qsing Pq#) we have: "G, with: + p#$# as the

classical inetic energ% of the particle with the original wave function # &e now

that this is also the classical inetic energ% of the time reversed particle, represented

b% T, in the T-transformed theor% T(=>"# 6ut the time differential term in ()" has

the behavior: (T)@t G -@t, so to obtain the correct result, we must deine the

classical kinetic ener"y operator "$or the tie reversed theoryb% T()" above,

instead of b% ()"# (he same result follows b% considering the wave 9#"

0imilarl%, using Pq#* we have: #Gp, wherep is the classical momentum (in

thex-direction" of the particle with the original wave function # &e now that this is

the ne"ativeclassical momentum of the time reversed particle, represented b% T#

he space differential term in (*" has the behavior: (T)@7 G @7, so to obtain

the correct result, we must deine the classical oentu operator #$or the tie

reversed theoryb% T(*" above, instead of b% (*"# (;gain, the same result follows b%

considering the wave 9, which must have the same momentum in the theor% 9(=>"

as in =>, but the differential term in (*" has the behavior: (9)@7 G -@7"#

hus, in the new theor%, T(=>", we find that the


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0ubstituted in T&)", this gives: )@)m )@7)G i @t, which is the T-reversal of

(1", as required#

Ret us now turn to the e% ob$ection to using T as the time reversal operator#

3. The key o!ection to using T"or Time Reversal.

he e% ob$ection is that Tdoes not have the right formal properties to properl%

represent time reversal in =>, whereas T*does# In particular, it is said that Tdoes not

transform energ% or momenta in the correct wa% for time reversal - for the energ% of

the reversed state of a particle must go be unchanged, whereas the momentum must

reverse, but (it is said" the use of T to transform to time reversed wave functions does

not satisf% this requirement# his point is repeated over and over again in different

forms# It is common in ordinar% te7t-boos, e#g#:

###we find that [the +eisenberg equation of motion] can be invariant onl% if

TT-+ -

his, however, is an unacceptable condition, because time reversal cannot change the

spectrum of, which consists of positive energies onl%# If Tis taen to be anti-unitar% [ie#

T*adopted], the 9 operator changes the i to -i [in the +eisenberg equation] and the trouble

does not occur# (SasiorowicB, [1], p#)"#

r >essiah:

&e are thus led to define a transformation of d%namical variables and d%namical states,

which we shall calltie reversal, in which rand ptransform respectivel% into rand -p# ### [

T9 ] obviousl% satisfies [the] relations# herefore we ma% tae [T9 ] as our time reversal

operator# ([1], p#//"#

;n e7panded version of this argument goes:

(i" he energ% of a time reversed wave function must be the same as the original

energ% M i#e# the time reversed spectrum must be the original, positive,

spectrum#

(ii" he energ% of a => wave function is given b%: "G i @t#

(iii" +owever, Thas the negative of the energ% of , for, b% (ii":

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"(T)G i (T)@t

G -i @t

G -",

contradicting (i"#

(iv" +ence Tcannot be the time reversal operator# It does not have the right formal

properties, since it reverses energ%, which time reversal cannot do#

(v" +owever, T*does uniquel% have the appropriate formal properties#

(vi" +ence T*is the onl% reasonable choice for time reversal#

he problem with this argument is in step (iii", because the law: "G i @t

represents the energ% in uantu echanics/but we are no longer considering

quantum mechanicsH we are transforming to thetie reversed theory, T&01). ;nd in

this theor%, the law for the energ% operator transforms to its reverse: "G -i @t#

ur wave function T is not a 01 (ave unction.It is not a => wave function

precisel% because its energ% equation is notthe => equation, as defined in (ii"# he

operator " in equation (ii" defines theclassical correlate o ener"y or a 01 (ave

unction. 0ince Tis not a => wave function, wh% should we assume that the

classical correlate of its energ% is defined in the same wa%N

0tep (iii" loos convincing, but it is circular,because it only applies under the

prior assuption that 01 is tie reversible,i#e#, that each time-reversed-is also a

=> wave function# n this assumption, then it would indeed be true that Tcannot be

the time reversal operator# I#e:I Tobeys 01 "iven that obeys 01, then T is not

the tie reversal operator, because "G -"(T"#+owever, this is equall% stated as

the converse fact: that i T is the tie reversal operator, then 01 is irreversible,

because "G -"(T"2 ;nd this gives the correct conclusion which is simpl% that T

is the tie reversal operator and 01 is irreversible#

he orthodo7 anal%sis has fallen into a peculiar ind of circular fallac% in

reasoning about this# &hat has not been recogniBed is the simple fact that when we

tae the transformed theoryT(=>", we find that we naturall% have to transform the

classical-quantum correspondence principles along with the wave functions

themselves, to obtain the empirical meaning of the theor%# I e7amine an e7plicit

treatment of this point ne7t, to illustrate how wea the orthodo7 argument is#

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#. Trans"orming the $lassical%Quantum $orrespondence &a's.

he previous ind of argument against T is drawn out at length b% 0achs [],

following &igner# he e% point of his argument is to distinguish the laws ()" and (*"

as


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formal interpretation of a ph%sical theor%# Uet he does not even tr% to give an%

$ustification for these general principles - he merel% states them as if the% are well-

nown facts# ;nd his procedure also contradicts the usual assumption that the concept

of time reversal is ob4ective: that the s%mmetr% itself is conceptuall% defined

independentl% of an% given theor%, and in anal%Bing the time reversal invariance of a

particular theor% we e7amine whether it ob$ectivel% satisfies this s%mmetr%# his

s%mmetr% is defined in all other branches of ph%sics as s%mmetr% under the

transformation: T: t-t,but0achs8 procedure instead requires us to define


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+e has provided his own summar% of mathematical arguments first put forward b%

&igner in 1'*)H but the conceptual basis for the underl%ing principles of time reversal

remain opaque#

he particular flaw in 0achs8 argument, and others of the same general ind, is

the assumption thatthe form of the


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r alternativel%:

(" (=>)("G i @t" 8iited 0uantiier

Pq#/ is read with ranging over all lo"ically possible (ave unctions. Interpreted this

wa%, it is a lo"ical deinitionof ", and we could simpl% rewrite " as a mathematical

operator:

(3" "G i @t

Pq# is read instead as sa%ing thator all (ave unctions, , that satisy 01, the

operator " has the propert% that: "G i @t# (&e can e7pand this more formall%

if we lie to: ()( %& ("G i @t"", but (" is easier to read"#

If the law ()" is intended as an empirical or theoretical law of =>, rather than

merel% as a definition, then this seems the natural reading#

+ow do we choose between these two readingsN It depends on whether we

intend to interpret " as merel% a


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(15" (=>)(@t G i @)m )@7)"

Aow ('" cannot possibl% be true, because there are wave functions that do not satisf%

('" M as we have seen with Tfor instance# !ather, (15" can be regarded as the

definition of the class => (for the limited simple theor% of free 1-dimensional wave

functions"# 6ut then, (15" b% itself maes no reference to an%thing empirical# If we

tae (15" to define the class of theoreticall% possible wave functions in quantum

mechanics, then we must engage additional laws lie Pq#)-, interpreted as epirical

la(s about easurable classical observables, to give us an empirical theor%#

Aote also that we cannot tae all three of Pq# )- as definitions# Eor suppose

we tae ) and * as definitions of the operators "and ## &e cannot also tae as a

deinition, because, given ()" and (*" are definitions, ("is siply not true o all

lo"ically possible s.In particular, it is easil% checed that (" is not true of the wave

function: T G ; e7p((i@ "(px% p#t@)m""#

Instead, what must be intended is that,"iven that ) and &3) are deinitions o "

and #, then the 01 (ave unctions obey &9). hen (" appears to be a contin"ent

proposition, which is true of uantu echanical (ave unctions,but not true in

general#

Siven this interpretation of Pqs# )-, it is obvious that ()" and (*" are invariant

under T, bein" siply atheatical deinitions,but (" is anti-s%mmetric under T.

his means that the classical commutation relations for the time reversed theor%,

T(=>", contradict the relations in the original theor%, => M naturall% enough,

because => is not T-invariant#

Uet this is e7actl% what 0achs8 argument denies M or rather, his argument

recognises that this would be the case if we adopted T for time reversal invariance, but

he denies that it is possible or the relation &9) to alter undertie reversal, and

concludes that T cannot represent tie reversal. Uet his onl% reason is that ()"-(" are


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affected b% this M we will find inevitabl% that => is not !I, but T is nonetheless a

perfectl% sensible transformation to appl% to the theor% =># &hat the interpretation

affects is the detailed view of the T transformations on the operators " and #, or on

the laws ()"-(" M since if we interpret these in two different wa%s, the% will naturall%

show different transformation properties#

4erhaps the most sensible interpretation, in line with ph%sicists8 normal practice,

is to tae ()"-(" to be intended to represent epirical la(s, and to tae their

transformations to be an alternative set of empirical laws# n this view, we can tae "

and # to be defined as mathematical operators, but we have to add something e7tra to

Pq#) and * to incorporate their contin"ent or epirical interpretation# &hat we add to

()" is an additional clause to the effect that:

()#P7tra" I "+ then (ould be the classically easured ener"y

o the particle represented by .

Aow having deined "b% ()", we can obtain its time reversed image, T"V b%

considering that T()" and ()" are both definitions or tautologies, and calculating:

T()" T&"G i @t"

(T"G T(i @t""

(T"G -(i @t""#

I#e# the definition of T" is:

T" G -(i

@t" G -"

0imilarl% for T#:

T(*" T(#G -i @7"

(T#G T&-i @7" "

(T#G -i @7 "

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I#e# the definition of T# is:

T# G -i @7 G #

here is no surprise in this# 6ut to obtain the T reversal of ()#P7tra", we have to

reason further: what would the time reversal of this empirical law stateN +ere we meet

a special and entrancing difficult%: there is no formal method of calculating such

reversals, because there is no formal method of representing contin"ency in ph%sics#

his will be somewhat m%sterious to ph%sicists, but it is quite clear to modern

intensional semanticists or logicians, because ph%sics has onl% an extensional

oralis, whereas the concept of contin"ency requires us to go to a deeper level of

intensional seantics, where we formal devices for quantif%ing over wave function after all, but b% the

corresponding T(=>" wave function 9#

hen when we measure a particle as having a classical energ%, we are not

measuring that: "Gfor =>H instead we are measuring b% an alternative

operator, "9, where: "9(9)G(9)for => and 9(=>"# 6ut then it

follows that, since "9(9)G(9)and "G, we must identif%: "9G -" G T"!

his is what is represented b% the law T()" above# 0imilar reasoning gives us

T(*", i#e# that: #9G -# G -T## he relation T(" follows similarl%#

+ence we obtain the natural time reversed theor%, T(=>", as Pq# T(1"-T(",

using the alternative operators "9and #9to represent the real empirical content, with

the implicit meaning that:

T()#P7tra" I "9+ then (ould be the classically easured ener"y

o the particle represented by .

1/


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;nd similarl% for the momentum relation#

&hat we have obtained as the theor% T(=>" is the natural wave functions that 0chrodinger originall% adopted, as illustrated in

Eig#1# he fact that this

is hardl% in dispute# It also seems natural that it correctl% represents the time reversed

image of ordinar% =># he arguments of 0achs and others that there is no coherent

theor% T(=>" are surel% mistaen# 6ut note the reason Callender gives in the second

quotation above is also not correct: time reversal does reverse the classical propert% of

momentumH the e% point is that it transforms the form of the classical operators to do

this#

+. ,ositivistic -rguments against T.

his brings us bac, however, to the first reason mentioned b% Callender in the

quotations above for adopting T* as the time reversal operator, and I will briefl%

comment on this# Pssentiall%, it now appears that the choice to use 01 rather than

T&01) is arbitrary, because there is no wa% of measuring the wave functions directl%,

and we cannot distinguish whether a wave function is and T(=>", are indistin"uishable in their

physical predictions, and so the% should be taen to represent the same theor%#

wo different inds of arguments should be distinguished here# he first M and

strongest - appeals to the


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do we dispense with the wave functions themselves, and %et retain the detailed

information the% represent that is required to predict interference effects correctl%N

0upposing this can be done, however, there is a second crucial point to be made

here: this view doesnot deny that T, i.e. the Racah operator, represents tie reversal

in uantu theory2!ather, it dissolves the physical dierence bet(een T and the

Wi"ner operator, T*, by iposin" aninterpretation o uantu echanics (here the

(aves and * are seen as physically identical ; or as odelin" the sae physical

reality. n this view, T should still be taen as the time reversal operator: the wor of

arriving at the conclusion that uantu echanics is nonetheless tie syetricis

reall% done b% the interpretation o the (ave unction# his alerts us to the fact that

whether (the deterministic part of" quantum mechanics is TRI is not solel% dependant

on adopting T,but also dependant on the interpretation o the physical reality o the

(ave unction. he orthodo7 account conflates these two points#

; second ind of argument, however, is based on a more general ind of

positivistic fallac%, which essentiall% involves arguing that the theories =>and

T(=>" are observationally indistin"uishable, and should thereore be re"arded as the

sae theory# ; version of the argument was given b% !eichenbach:#

here remains the problem of distinguishing between (q,t" and (q,-t"# In order to

discriminate between these two functions, we would first have to now whether [PG i

@t or: PG -i @t ] is the correct equation# 6ut the sign on the right in 0chrodingerFs

equation can be tested observationall% onl% if a direction of time has been previousl% defined#

&e use here the time direction of the macroscopic s%stems b% the help of which we compare

the mathematical consequences of 0chrodingerFs equation with observation# herefore, to

attempt a definition of time direction through 0chrodingerFs equation would be reasoning in a

circleH this equation merel% presents us with the time direction we introduced previousl% in

terms of macrocosmic processes# ([1], pp# )5'-)15"#

Aote that !eichenbach does notden% that: T(q,t" G (q,-t" does in fact represent

time reversal# Instead, he argues that the wave functions: (q,t" and T(q,t" are

observationall% indistinguishable, and that this undermines the conclusion that => is

irreversible#

; number of confusions can be found here, but there is one e% point that needs

to be made# 0uppose that we have a time as%mmetric theor%, T, which we regard as

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observationall% indistinguishable from its reversal, TT! 4ositivistic reasoning suggests

that T andTT are therefore identical theories# 6ut we can construct a weaer theor%:

(Tor TT"V i#e# the dis$unction of T and its time reversal, TT, and this weaer theor% is

obviousl% time s%mmetric# If we suppose that T and TTare logicall% equivalent M or

have identical meanings - this would entail that T is identical to the theor% (Tor TT"#

6ut this cannot be correct# Certainl%, T entails (Tor TT", because an%

observational evidence for Tis also evidence for: (Tor TT"# n the other hand, can

we have evidence that T is true, whereas TT is not true, so that we can establish the

stronger, non-TRItheor% Tb% itselfN he positivist account maes Tand TT appear

indistinguishable: for how can we distinguish whether we are really in a T-universe,

or in a TT-universeN ;ssuming we cannot, we are led to tae Tto be equivalent to: T

or TT# 6ut this is not a valid argument: instead it is an illustration of a t%pical ind of

fallac% inherent in positivist-empiricist conceptions of meaning# (Indeed,

!eichenbach8s reasoning would remove the possibilit% of ever establishing a time

as%mmetric theor%, because we could reduce an% theor% T to: (T or TT"#" 0uch

fallacies have deepl% infected the sub$ect, and the% are onl% properl% dispelled b%

tacling modern semantics seriousl%#

his is not the place to anal%se such fallacies, but there is a ver% important

application to the present problem, which brings us bac to the view of Costa de

6eauregard, who argues that T is indeed the correct time reversal operator in

relativistic uantu echanics ; but that this theory is nonetheless TRI or reversible

&unlike the non-relativistic theory).

Aote that the dis$unction: (=> or T(=>" " should be read with the

quantification:

(=> or T(=>"" ( that are ph%sicall% real)(=>" or

( that are ph%sicall% real)(T(=>"

Aow both => and T(=>" taen separatel% do entail something ver% significant: that

all is not: (=> or T(=>"", but the

weaer dis$unction, which I will call =>;:

1'


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=>; ()(=> or T(=>" "

=>; allows mi7tures and superpositions of wave functions with opposite directions

of comple7 rotation# =>; contradicts both =>, and (=> or T(=>"", and is trul%

!I#

Aow what is the evidence that =>, or (=> or (=>"" is true, rather than

=>;N It is the observation that ordinar% particles alwa%s have a coonrelative

tie orientation. 6ut this onl% appears necessar% in non-relativistic =># he

interpretation of the T operation in non-relativistic quantum theor% leaves particle

t%pes (e#g# electrons" as the same particle t%pes (electrons", and onl% transforms the

tra$ectories (not the charges", and in this case, =>; cannot be realistic M because we

cannot have one electron that satisfies => and another electron that satisfies T(=>"#

Costa de 6eauregard8s suggestion [11] means that if we tae T to transform particles

(e#g# electrons" into their anti-particles (positrons", with anti-particles having the

opposite direction of comple7 rotation, then the theor% will have the form of =>;

after all M since anti-particles do e7ist# &hat is wrong with taing electrons to satisf%

=>, andpositrons to satisf% T(=>" and T to transform the char"es of particles, sothat electrons transform to positrons M as Ee%nmann8s interpretation suggestsN

If de 6eauregard8s arguments are correct, the fundamental transformations for

the time reversal, charge reversal, and space reversal have been misconstrued# 6ut

while his arguments support the idea that T rather than T* is the time reversal

operator, the conclusion is that relativistic uantu echanics is nonetheless TRI, or

reversible, because the correct interpretation is lie =>;, which has a quite different

logical structure to =># 6ut further discussion of this point of view is be%ond the

scope of this paper#

. $onclusions.

he orthodo7 account of time reversal transformations in quantum theor% presented

authoritativel% in a wide range of te7tboos and specialiBed treatises is conceptuall%

inadequate# he arguments t%picall% put forward that T* rather than T must be

adopted as the time reversal operator in quantum mechanics for logical reasons are

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mistaen# here is no reason to re$ect the T operator on such grounds# ;lternative

positivist arguments from the and T(=>" are also laden

with errors# ;rguments from the in this respect is not a

ph%sical feature of the universe -because this interpretation denies that the wave

function is itself ph%sical#

hese conceptual flaws in the account of time s%mmetr% of quantum theor%,

when considered along with the decisive flaws in the account of time s%mmetr% of the

probabilistic component of quantum theor% raised b% &atanabe, +eale%, 4enrose,

Callender, and +olster, should be a cause for deep concern# he% show how poorl%

the conceptual foundations of quantum theor% are understood# If there is an% single

culprit for this state of affairs, it is the complacenc% engendered b% the positivist

approach to conceptual anal%sis in ph%sics# Eor despite being accepted as deepl%

inadequate b% philosophers and logicians for over fift% %ears, positivism unfortunatel%

remains as the central point of departure in man% conceptual accounts of quantum

ph%sics, and is found at the center of the orthodo7 anal%ses of the sub$ect of time

reversal#

T is indeed the time reversal operator in ordinar% quantum mechanics, and this

theor% fails to be time reversal invariant unless we adopt the probabilistic

interpretation# +owever the arguments of Costa de 6eauregard show that relativistic

quantum theor% ma% be convincingl% interpreted as being !I b% adopting the

Ee%nmann interpretation of anti-particles as


22/23

Re"erences.

[1] Callender, C# Is ime +anded in a =uantum &orldN?#=roc.'rist.>oc, '

()555" pp )-)/'#

[)] &atanabe, 0atosi# X0%mmetr% of 4h%sical Raws# 4art *# 4rediction and

!etrodiction#XRev.1od.=hys.'((1" (1'" pp 1'-13/#

[*] &atanabe, 0atosi# XConditional 4robabilit% in 4h%sicsX# >uppl.=ro".Theor.=hys.

&5yoto) xtra ?uber (1'/" pp 1*-1/#

[] +eale%, !# 0tatistical heories, =uantum >echanics and the .irectedness of

ime?# (1'31", pp#''-1)# InReduction, Tie and Reality, ed# !# +eale%#

(Cambridge: Cambridge Qniversit% 4ress# 1'31"#

[] +olster, ;## he criterion for time s%mmetr% of probabilistic theories and the

reversibilit% of quantum mechanics?# ()55*"#?e( @ournal o =hysics

(www#n$p#org", ct# )55*#http:@@stacs#iop#org@1*/-)/*5@@1*5#

[/] .avies, 4#C# The =hysics o Tie 'syetry.(QO: 0urre% Qniversit% 4ress#

1'#"

[] 0achs, !obert S# The =hysics o Tie Reversal.(Chicago: Qniversit% of

Chicago 4ress# 1'3#"

[3] 2eh, +#.# The =hysical Aasis o the Birection o Tie.(6erlin: 0pringer-Kerlag#

1'3'#"

['] 0chrodinger, P# XIrreversibilit%X#=roc.Royal Irish 'cadey)* (1'5" pp 13'-

1'#

[15] 4enrose, !# The peroreulen, ;lice#' andbook o 8o"ic and

8an"ua"e.(Cambridge, >assachusetts: he >I 4ress# 1''#"

))
http://www.njp.org/http://stacks.iop.org/1367-2630/5/130http://stacks.iop.org/1367-2630/5/130http://www.njp.org/http://stacks.iop.org/1367-2630/5/130


23/23

[1] !eichenbach, +# The Birection o Tie.(6erele%: Qniversit% of California

4ress# 1'/#"

The Quantum Mechanical Time Reversal Operator

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