EXAMPLES OF QUANTUM INTEGRALS

EXAMPLES OF QUANTUM INTEGRALS Stn Gudder Deprtment of Mthemtics University of Denver Denver, Colordo 88 sgudder@mth.du.edu Abstrct We first consider method of centering nd chnge of vrible formul for quntum integrl. We then present three types of quntum integrls. The first considers the expecttion of the number of heds in n flips of quntum coin. The next computes quntum integrls for destructive pirs exmples. The lst computes quntum integrls for (Lebesgue) quntum mesure. For this lst type we prove some quntum counterprts of the fundmentl theorem of clculus. Introduction Quntum mesure theory ws introduced by R. Sorkin in his studies of the histories pproch to quntum mechnics nd quntum grvity [7]. Since then, he nd severl other uthors hve continued this study [,, 6, 8, 9, ] nd the uthor hs developed generl quntum mesure theory for infinite crdinlity spces [3]. Very recently the uthor hs introduced the concept of quntum integrl [4]. Although this integrl generlizes the clssicl Lebesgue integrl, it my exhibit unusul behviors tht the Lebesgue integrl does not. For exmple, the quntum integrl my be nonliner nd nonmonotone. Becuse of this possible nonstndrd behvior we lck intuition concerning properties of the quntum integrl. To help us gin some intuition for this new integrl, we present vrious exmples of quntum integrls.

The pper begins with method of centering nd chnge of vrible formul for quntum integrl. Exmples of centering nd vrible chnges re given. We lso consider quntum integrls over subsets of the mesure spces. We then present three types of quntum integrls. The first considers the expecttion n of the number of heds in n flips of quntum coin. We prove tht n symptoticlly pproches the clssicl vlue n/ s n pproches infinity, nd numericl dt re given to illustrte this. The next computes quntum integrls for destructive pirs exmples. The functions integrted in these exmples re monomils. The lst computes quntum integrls for (Lebesgue) quntum mesure. For this lst type, some quntum counterprts of the fundmentl theorem of clculus re proved. Centering nd Chnge of Vribles If (X, A) is mesurble spce, mp µ: A R is grde- dditive [,, 3, 7], if µ (A B C) µ (A B) µ (A C) µ (B C) µ(a) µ(b) µ(c) for ny mutully disjoint A, B, C A where A B denotes A B whenever A B. A q-mesure is grde- dditive mp µ: A R tht lso stisfies the following continuity conditions [3] (C) For ny incresing sequence A i A we hve µ ( A i ) lim i µ(a i ) (C) For ny decresing sequence B i A we hve µ ( A i ) lim i µ(a i ) A q-mesure µ is not lwys dditive, tht is, µ (A B) µ(a)µ(b) in generl. A q-mesure spce is triple (X, A, µ) where (X, A) is mesurble spce nd µ: A R is q-mesure [, 3, 7]. Let (X, A, µ) be q-mesure spce nd let f : X R be mesurble function. It is shown in [4] tht the following rel-vlued functions of λ R re Lebesgue mesurble: λ µ ({x: f(x) > λ}), λ µ ({x: f(x) < λ})

We define the quntum integrl of f to be fdµ µ ({x: f(x) > λ}) dλ µ ({x: f(x) < λ}) dλ (.) where dλ is Lebesgue mesure on R. If µ is n ordinry mesure (tht is; µ is dditive) then fdµ is the usul Lebesgue integrl [4]. The quntum integrl need not be liner or monotone. Tht is, (f g)dµ fdµ gdµ nd fdµ gdµ whenever f g, in generl. However, the integrl is homogenious in the sense tht αfdµ α fdµ, for α R. Definition (.) gives the number zero specil sttus which is unimportnt when µ is mesure, but which mkes nontrivil difference when µ is generl q-mesure. It my be useful in pplictions to define for R the -centered quntum integrl fdµ µ [ f (λ, ) ] dλ µ [ f (, λ) ] dλ µ [ f (λ, ) ] dλ µ [ f (, λ) ] dλ (.) Of course, fdµ fdµ but we shll omit the subscript. Our first result shows how to compute fdµ when f is simple function. Lemm.. Let f n i α iχ Ai be simple function with A i A j, i j, n ia i X. If α < α m < α m < < α n then ( n ) ( n ) fdµ (α m )µ A i (α m α m )µ A i im im [ ( m ) (α n α n )µ(a n ) ( α m )µ A i (α m α m )µ ( m i i A i ) (α α )µ(a ) ] (.3) 3

α µ(a ) α [µ (A A ) µ(a )] α m [µ (A A m ) µ (A m A m ) µ(a ) µ(a m ) (m )µ(a m )] α m [µ (A m A m ) µ (A m A n ) (n m )µ(a m ) µ(a m ) µ(a n )] α n [µ (A n A n ) µ(a n )] α n µ(a n ) [ ( m ) ( n )] µ A i µ A i i im (.4) Proof. Eqution (.3) is strightforwrd ppliction of the definition (.). We cn rewrite (.3) s ( n ) ( n )] fdµ α m [µ A i µ A i α m [µ im ( n im A i ) µ im ( n im3 A i )] α n [µ (A n A n ) µ(a n )] α n µ(a n ) α µ(a ) ( ) ] ( 3 ) ( )] α [µ A i µ(a α 3 [µ A i µ A i i ( m ) α m [µ A i µ i [ ( m ) ( n µ A i µ i im Applying Theorem 3.3 [3] the result follows. ( m i A i )] i A i )] Corollry.. If µ is mesure nd f is integrble, then fdµ fdµ µ(x) Proof. By (.4) the formul holds for simple functions. Approximte f by n incresing sequence of simple functions nd pply the monotone convergence theorem. 4 i

As n illustrtion of Lemm., let f χ A bχ B be simple function with A B, A B X, < b. By (.4) we hve fdµ [µ (A B) µ(b)] bµ(b) This lso shows tht the quntum integrl is nonliner becuse if µ (A B) µ(a) µ(b) then (χ A bχ B )dµ fdµ µ(a) bµ(b) χ A dµ b χ B dµ Corollry. shows tht if µ is mesure, then fdµ is just trnsltion of fdµ by the constnt µ(x) for ll integrble f. We now show tht this does not hold when µ is generl q-mesure. Exmple. Let > be fixed constnt nd let f cχ A be simple function with c nd A, X. We cn write f in the cnonicl form f χ A cχ A where A is the complement of A. By Lemm. we hve tht fdµ cµ(a) nd pplying (.4) to the vrious cses we obtin the following: Cse. If < < c, then α, α c, α < < α, A A nd A A. We compute fdµ µ(a ) cµ(a) [µ(a ) µ(a)] fdµ [µ(a ) µ(a)] Cse. If < c <, then α, α c, α < α <, A A nd A A. We compute fdµ µ(a ) c [µ(x) µ(a )] µ(x) fdµ c [µ(a) µ(a ) µ(x)] µ(x) Cse 3. If c < <, then α c, α, α < α <, A A nd A A. We compute fdµ cµ(a) [µ(x) µ(a)] µ(x) fdµ µ(x) 5

We now derive chnge of vrible formul. Suppose g is n incresing nd differentible function on R nd let g (± ) lim λ ± g (λ). If f : X R is mesurble, then so is g f nd we hve g fdµ µ [{x: g f(x) > λ}] dλ µ [{x: g f(x) < λ}] dλ µ [{ x: f(x) > g (λ) }] dλ [{ x: f(x) < g (λ) }] dλ Letting t g (λ), g(t) λ, g (t)dt dλ, by the usul chnge of vrible formul we obtin g fdµ (.5) g ( ) g () µ [{x: f(x) > t}] g (t)dt g () g ( ) µ [{x: f(x) < t}] g (t)dt For exmple, if f, letting g(t) t n we hve f n dµ µ [{x: f(x) > t}] nt n dt (.6) A As with the Lebesgue integrl, if A A we define fdµ χ A dµ We then hve fdµ µ [{x: χ A (x)f(x) > λ}] dλ A µ [ A f (λ, ) ] dλ µ [{x: χ A (x)f(x) < λ}] dλ µ [ A f (, λ) ] dλ { µ [ A f (λ, ) ] µ [ A f (, λ) ]} dλ Now (A, A A) is mesurble spce nd it is esy to check tht µ A (B) µ(a B) is q-mesure on A A so (A, A A, µ A ) is q-mesure spce. Hence, for mesurble function f : X R, the restriction f A: A R is mesurble nd fdµ f Adµ A A Similr definitions pply to the centered integrls A fdµ. 6

3 A Quntum Coin If we flip coin n times the resulting smple spce X n consists of n outcomes ech being sequence of n heds or tils. For exmple, possible outcome for 3 flips is HHT nd X {HH, HT, T H, T T }. If this were n ordinry fir coin then the probbility of subset A X n would be A / n where A is the crdinlity of A. However, suppose we re flipping quntum coin for which the probbility is replced by the q-mesure µ n (A) A / n. It is esy to check tht µ is indeed q-mesure. In fct the squre of ny mesure is q-mesure. Let f n : X n R be the rndom vrible tht gives the number of heds in n flips. For exmple, f 3 (HHT ). For n ordinry coin the expecttion of f n is n/. We re interested in computing the quntum expecttion fn dµ n for quntum coin. For the cse n we hve X {x, x } with f (x ), f (x ). Then f χ {x } nd by (.3) we hve f dµ µ ({x }) 4 For the cse n, we hve X {x, x, x 3, x 4 } with f (x ), f (x ) f (x 3 ), f (x 4 ). Then f χ {x,x 3 } χ {x } nd by (.3) we hve f dµ µ ({x, x, x 3 }) µ ({x }) 9 6 6 5 8 Continuing this process, X 3 {x,..., x 8 } nd f χ {x5,x 6,x 7 } χ {x,x 3,x 4 } 3χ {x } By (.3) we obtin f 3 dµ 3 µ 3 ({x,..., x 7 }) µ 3 ({x,..., x 4 }) µ 3 ({x }) 49 64 6 64 64 33 3 7

For 4 flips, X 4 {x,..., x 6 } nd f 4 dµ 4 µ 4 ({x,..., x 5 }) µ 4 ({x,..., x }) µ 4 ({x,..., x 5 }) µ ({x }) 5 5 93 6 64 For 5 flips, X 5 {x,..., x 3 } nd f 5 dµ 5 µ 5 ({x,..., x 3 }) µ 5 ({x,..., x 6 }) µ 5 ({x,..., x 6 }) µ 5 ({x,..., x 6 }) µ ({x }) 3 6 6 6 965 3 5 Letting n f n dµ n we hve tht ( ) 4 { ( ) 3 6 { (3 ). n n { (n ) [( ) [( ) 3 [( n ) ( )] } ( )] 3 [( ) 3 ( )] n ( ) 3 [( ) n ( )] } 3 ( ) ( )] } n n n We shll show tht n symptoticlly pproches the clssicl vlue n/ for lrge n; tht is n lim n n (3.) As numericl evidence for this result the first seven vlues of n /n re:.5,.65,.6875,.766,.7539,.7749,.795 nd the twentieth vlue is.8737. The next result shows tht the quntum expecttion does not exceed the clssicl expecttion. 8

Lemm 3.. For ll n N, f n dµ n n/. Proof. Letting A i fn ({i}), i,..., n, pplying (.4) we obtin f n dµ n [µ n (A A ) µ n (A A n ) (n )µ n (A ) µ n (A ) µ n (A n )] [µ n (A A 3 ) µ n (A A n ) (n 3)µ n (A ) µ n (A 3 ) µ n (A n )]. (n )[µ n (A n A n ) µ n (A n )] nµ n (A n ) { A n A A A n (n ) A A A n [ A A 3 A A n (n 3) A A 3 A n ]. (n )[ A n A n A n ] n A n } n { A A ( A A n ) [ A A ( A 3 A n ) ] (n ) [ A n A n A n ] n A n } By the binomil theorem we conclude tht f n dµ n n { A [ ( n A )] A [ ( n A A )] (n ) A n [ ( n A n A )] n A n [ n A A A n ]} n [ A n A n n A n n ] n ( A A n A n ) n where the lst equlity follows from the clssicl expecttion. 9

We now give n in closed form nd prove (3.). Theorem 3.. () For ll n N we hve [ (b) Eqution (3.) holds. Proof. () Letting b n ( ) n [( ) n n n ( n ( n n ( )] n ) )] n [( ) n it is shown in [5] tht b n (n ) n ( ) n n n ( ) ( )] n n n Since n b n / n, the result follows. (b) By Stirling s formul we hve the symptotic pproximtion for lrge n. Hence, lim n n n! πn nn e n ( ) n (n)! lim n n n (n!) lim n lim n πn πn (n) n n e n e n πnn n Hence, n lim n n ( n lim n n ) lim n ( n n ) n The next exmple illustrtes the -centered integrl f dµ for two flips of quntum coin.

Exmple. The following computtions result from pplying (.3). If, then f dµ ( )µ ({x, x, x 3, x 4 }) µ ({x, x, x 3 }) µ ({x }) 5 8 If, then f dµ ( )µ ({x, x, x 3 }) µ ({x }) ( )µ ({x 4 }) ( ) 9 6 6 6 5 8 5 8 If, then f dµ ( )µ ({x }) ( )µ ({x, x 3, x 4 }) µ ({x 4 }) ( ) 6 ( ) 9 6 6 5 8 5 8 If, then f dµ [( )µ ({x, x, x 3, x 4 }) µ ({x, x 3, x 4 }) µ ({x 4 })] [ ( ) 9 6 ] 6 8 We conclude tht f dµ n is piecewise liner s function of. 4 Destructive Pirs Exmples Consider X [, ] s consisting of prticles for which pirs of the form (x, x 3/4), x [, /4] re destructive pirs (or prticle-ntiprticle pirs). Thus, prticles in x [, /4] nnihilte their counterprts in [3/4, ] while prticles in (/4, 3/4) do not interct with other prticles. Let B(X) be the set of Borel subsets of X nd let ν be Lebesgue mesure on B(X). For A B(X) define µ(a) ν(a) ν ({x A: x 3/4 A})

Thus, µ(a) is the Lebesgue mesure of A fter the destructive pirs in A nnihilte ech other. For exmple, µ ([, ]) / nd µ ([, 3/4]) 3/4. It cn be shown tht (X, B(X), µ) is q-mesure spce [3]. Letting f(x) x nd < b we shll compute We first define f(x)dµ [,b] f(x)dµ F (λ) µ ({ x: fχ [,b] (x) > λ }) µ ({x [, b] : x > λ}) µ ((λ, b]) If b 3/4 then We obtin xdµ F (λ) { b λ if λ b if λ > b F (λ)dλ (b λ)dλ b which is the expected clssicl result becuse there is no interference (nnihiltion). Now suppose tht b 3/4 in which cse there is interference. If λ b 3/4, then F (λ) b λ s before. If λ < b 3/4, then F (λ) (b λ) (b 34 ) λ 3 λ b We then obtin xdµ F (λ)dλ 3 b 9 6 b 3/4 ( ) 3 b λ b dλ (b λ)dλ b 3/4 For exmple, xdµ 7/6 which is slightly less tht xdλ /. Of course, interference is the cuse of this difference. Also, 3/4 xdµ 9/3 which grees with 3/4 xdx s shown in the b 3/4 cse.

We next compute xn dµ. If b 3/4, then by our chnge of vrible formul we hve x n dµ n (b λ)λ n dλ bn n in greement with the clssicl result. If b 3/4, we obtin by the chnge of vrible formul x n dµ n F (λ)λ n dλ [ 3/4 ( b n 3 λ b) λ n dλ n [ b n ( ) ] b 3 n 4 b 3/4 (b λ)λ n dλ As check, if n we obtin our previous result. Notice tht the devition from the clssicl integrl becomes x n dx x n dµ n ( ) b 3 n 4 which increses s b pproches. We now chnge the previous exmple so tht we only hve destructive pirs in which cse we obtin more interference. We gin let X [, ], but now we define the q-mesure µ(a) ν(a) ν ({ x A: x A}) In this cse, (x, x /), x [, /] re destructive pirs. For instnce, µ(x), µ ([/6, 5/6]) /3, nd µ ([, /]) /. Letting f(x) x nd < b, we shll compute xdµ xdµ We then hve (,b) F (λ) µ ({ x: fχ (,b) (x) > λ }) µ ({x (, b): x > λ}) µ ((, b)) if λ µ ((λ, b)) if λ b if λ b ] 3

Now { x (, b): x (, b)} if nd only if b /. If b / we hve b if λ F (λ) b λ if λ b if λ b We then obtin xdµ (b )dλ (b λ)dλ b which is expected becuse there is not interference. If b /, letting c b / we hve tht c nd µ ((, b)) b (c ) b ( b ) b If λ b /, then µ ((λ, b)) λ b nd if λ b /, then µ ((λ, b)) b λ. Hence, b if λ F (λ) λ b if λ b / b λ if b / λ b We conclude tht xdµ ( b )dλ b b 4 / (λ b )dλ The devition from the clssicl integrl becomes xdx xdµ b b 4 b / (b λ)dλ 4

Notice tht if nd only if b /. Specil cses of the integrl re / xdµ b b 4 xdµ 8 3/4 xdµ 7 3 xdµ 4 5 (Lebesgue) Quntum Mesure We gin let X [, ] nd let ν be Lebesgue mesure on B(X). We define (Lebesgue) q-mesure by µ(a) ν(a) for A B(X) nd consider the q-mesure spce (X, B(X), µ). The first exmple in this section is the - centered quntum integrl x n dµ. Applying the chnge of vrible formul we obtin x n dµ n µ ({x: x > t}) t n dt n µ ({x: x < t}) t n dt n ( t) t n dt n ( (n )(n ) n t t n dt n n n ) n As specil cses we hve xdµ 3 3 3 x n dµ (n )(n ) We now compute the quntum integrl xn dµ for < b. Agin 5

the chnge of vrible formul gives x n dµ n n As specil cses we hve µ ( (, b) { x: x > λ /n}) dλ µ ((, b) {x: x > t}) t n dλ (b t) t n dt n (b ) t n dt (n )(n ) (bn n ) n (b ) n xdµ b3 3 3 3 (b ) dµ (b ) We cn compute xn dµ nother wy without relying on chnge of vribles: x n dµ µ ( (, b) { x: x > λ /n}) dλ n n (b λ /n ) dλ n n n (b ) dλ (b bλ /n λ /n )dλ (b ) n (n )(n ) bn n ( b n ) n which grees with our previous result. Until now we hve only integrted monomils. We now integrte the more complex function e x. By the chnge of vrible formul e x dµ µ ((, b) {x: x > t}) e t dt (b t) e t dt (b ) e t dt [ e b e e (b ) ] 6

In prticulr, e x dx (e b b) For the Lebesgue integrl we hve the formul f(x)dx f(x)dx f(x)dx which is frequently used to simplify computtions. This formul does not hold for our q-mesure µ. However, we do hve the following result. Theorem 5.. If f is incresing, differentible, nonnegtive on [, ] nd f ( ) b, f (), then fdµ fdµ fdµ (b ) Proof. Employing the chnge of vrible formul gives fdµ f ( ) f () µ ((, b) {x: x > t}) f (t)dt (b t) f (t)dt (b t) f (t)dt f () (b ) f (t)dt On the other hnd, using integrtion by prts we hve fdµ fdµ (b t) f (t)dt b f() (b t) f (t)dt f(t)dt (b t) f (t)dt (b ) f() (b t) f (t)dt ( t) f (t)dt f() [ (b t) ( t) ] f (t)dt (b )f() (b t) f (t)dt (b )f() (b ) 7 (b t) f (t)dt tf (t)dt

The result now follows. (b t) f (t)dt (b ) f() (b ) (b t) f (t)dt f(t)dt Exmple 3. In this exmple we use some previous computtions to verify Theorem 5.. We hve shown tht x n dµ Hence, by Theorem 5. we hve x n dµ x n dµ (n )(n ) bn x n dµ (b ) t n dt (n )(n ) (bn n ) n (b ) n which grees with our previous result. We hve shown tht Hence, by Theorem 5. we hve e x dµ e x dµ e x dµ (e b b) e x dµ (b ) e t dt (e b b) (e ) (b )(e ) [ e b e e (b ) ] which grees with our previous result. Exmple 4. We compute the quntum integrl of f(x) x x. By the chnge of vrible formul we hve (x x )dµ b3 3 b4 6 (b t) ( t)dt (b t) dt (b t) tdt 8

This gives the surprising result tht (x x )dµ xdµ x dµ We shll lter show tht this quntum integrl is lwys dditive for sums of incresing continuous functions even if they re not differentible. The next exmple shows tht this result does not hold for two monomils if their sum is not incresing. Exmple 5. Let f(x) x x for x [, ]. To evlute f(x)dµ we cnnot use the chnge of vrible formul becuse f is not incresing, so we will proceed directly. Let / b. For λ /4 we hve tht λ x x, if nd only if x ( ± 4λ ) /. Hence, for λ b b we hve ν ( (, b) { x: x x > λ }) { 4λ if λ 4 if /4 λ nd for λ b b we hve Hence, ν ( (, b) { x: x x > λ }) b 4λ (x x )dµ b ( b ) /4 4λ dλ ( 4λ)dλ b b 4 3 b b 5 3 b3 5 6 b4 Notice tht this does not coincide with xdµ x dµ 3 b3 6 b4 For completeness we evlute the integrl with b /. Since f is incresing on this intervl we obtin the expected result: (x x )dµ b ( b 4λ ) dλ 3 b3 6 b4 9

Exmple 6. Let f be the following piecewise liner function: { x if x / f(x) x if / x Let / b. For λ b we hve nd for b λ we hve Hence fdµ b ν ((, b) {x: f(x) > λ}) b λ ν ((, b) {x: f(x) > λ}) λ ( ) b λ dλ ( λ) dλ b 3 4b b 3 b If b / we obtin the expected result Observe tht fdµ b d db d db ( ) b λ dλ 3 b3 x n dµ b n e x dµ e b However, in Exmple 5 we hve for b > / tht d db (x x )dµ 5b 5b 3 b b nd in Exmple 6 we hve for b > / tht d db fdµ 4 6b b f(b) These exmples gin illustrte the specil nture of incresing functions. The next result is quntum counterprt to the fundmentl theorem of clculus.

Theorem 5.. If f is continuous nd monotone on [, ], then d db fdµ f(b) Proof. If f is decresing then f is incresing so we cn ssume f is incresing. For positive integer n, let g be the following incresing step function on [, ]: g c χ [,/n] c χ (/n,/n] c n χ ((n )/n,] where < c < < c n. For < b we hve tht (m )/n < b m/n for some integer < m n nd gχ [,b] c χ [,/n] c χ (/n,/n] c m χ ((m )/n,(m )/n] c m χ ((m )/n,b] Letting A i ((i )/n, i/n], i,..., m, A m ((m )/n, b] nd b b (m )/n we hve by (.4) of Lemm. tht gdµ c [µ (A A ) µ (A A m ) (m )µ(a ) µ(a ) µ(a m )] c [µ (A A 3 ) µ (A A m ) (m 3)µ(A ) µ(a 3 ) µ(a m )] c m [µ (A m A m ) µ(a m )] c m µ(a m ) ( ) ( ) ( ) ] c [(m ) n n b (m 4) n b ( ) ( ) ( ) ] c [(m 3) n n b (m 6) n b [ ( ) ] c m n b b c m b c [(m 3) n ] [ n b c (m 5) n ] n b ( c m n ) n b c m b It follows tht d db gdµ n (c c c m ) c m b (5.)

nd tht d db gdµ c m g(b) (5.) We cn ssume without loss of generlity tht f is nonnegtive. Then there exists n incresing sequence of incresing nonnegtive step functions s i converging uniformly to f. Since µ [ s i (λ, )] µ [ s i (λ, ) ] we hve by the continuity of µ tht µ [ f (λ, ) ] µ [ s i (λ, ) ] lim µ [ s i (λ, ) ] These sme formuls pply to fχ [,b] nd s i χ [,b]. By the quntum bounded monotone convergence theorem [4] we conclude tht fdµ i lim s i dµ Applying (5.) with g replced by s i it cn be checked tht the sequence of functions of b given by d b s i dµ db is uniformly Cuchy so the sequence converges nd hence By (5.) d db converges uniformly so we hve d db fdµ lim d db d db s i dµ d fdµ lim db s i dµ s i dµ f(b) Lemm 5.3. If f is continuous nd monotone on [, ], then [ d b ] fdµ () db

Proof. We cn ssume without loss of generlity tht f is incresing. Let g c i χ Ai be step function s in the proof of Theorem 5.. If b is sufficiently smll we hve gχ [,b] c χ A [,b] c χ [,b] Hence, for such b we hve gdµ gχ [,b] dµ c χ [,b] dµ c b Therefore, [ d db ] ( ) d gdµ () db c b () As shown in the proof of Theorem 5., there exists n incresing sequence of step functions s i such tht The result follows. d db fdµ lim d db s i dµ Prt (b) of the next theorem is the second hlf of the quntum fundmentl theorem of clculus. Theorem 5.4. () If f is continuous nd monotone on [, ], then fdµ t f(x)dxdt (b) If f is monotone nd continuous on [, ], then f dµ f(b) f() f ()b Proof. () If g f, then integrting gives t f(x)dx g (t) g () 3

Integrting gin we hve t Hence, for ll b [, ] we hve Since by Theorem 5. letting g(b) g (). Hence, f(x)dxdt g(b) g() g ()b g(b) g() g ()b t d fdµ f(b) db f(x)dxdt fdµ we hve tht g() nd by Lemm 5.3 we obtin t (b) By Prt () we hve f dµ f(x)dxdt g(b) t f (x)dxdt f(b) f() f ()b The next corollry follows from Theorem 5.4() fdµ [f (t) f ()] dt Corollry 5.5. The quntum (Lebesgue) integrl is dditive for incresing (decresing) continuous functions. Exmple 7. We compute some quntum integrls using Theorem 5.4(). cos xdµ sin xdµ cosh xdµ t t t cosh b cos xdxdt sin xdxdt cosh xdxdt sin tdt ( cos b) ( cost)dt (b sin b) sinh tdt The lst integrl shows tht the quntum counterprt of e x is cosh x. 4

Acknowledgement. The uthor thnks Petr Vojtěchovský for pointing out reference [5] References [] Y. Ghzi-Tbtbi, Quntum mesure theory: new interprettion, rxiv: qunt-ph (96.94), 9. [] S. Gudder, Finite quntum mesure spces, Amer. Mth. Monthly (to pper). [3] S. Gudder, Quntum mesure theory, Mth. Slovc (to pper) nd http://www.mth.du.edu/preprints.html. [4] S. Gudder, Quntum mesure nd integrtion theory, rxiv: quntph(99.3), 9 nd http://www.mth.du.edu/preprints.html. [5] M. Hirschhorn, Clkin s binomil identity, Discrete Mth. 59 (996), 73 78. [6] R. Slgdo, Some identities for the quntum mesure nd its generliztions, Mod. Phys. Letts. A 7 (), 7 78. [7] R. Sorkin, Quntum mechnics s quntum mesure theory, Mod. Phys. Letts. A 9 (994), 39 37. [8] R. Sorkin, Quntum mechnics without the wve function, J. Phys. A 4 (7), 37 33. [9] R. Sorkin, An exercise in nhomomorphic logic, J. Phys. A (to pper). [] S. Sury nd P. Wllden, Quntum covers in quntum mesure theory, rxiv: 89.95 [qunt-ph] (8). 5