Feynman s path integral approach to quantum physics and its relativistic generalization

Transcription

1 Feynman s path integral approach to quantum physics and its relativistic generalization Jürgen Struckmeier j.struckmeier@gsi.de, struck Vortrag im Rahmen des Winterseminars Aktuelle Probleme der Beschleuniger- und Plasmaphysik des Instituts für Angewandte Physik der Johann Wolfgang Goethe-Universität Frankfurt am Main Riezlern, März 2009 Feynman s path integral p. 1

2 Outline Classical action principle Canonical quantization Quantum action principle Classical limit Riemann integral, path integral Example 1: Transition amplitude for a free particle Example 2: Derivation of the Schrödinger equation Relativistic generalization of the path integral Conclusions Feynman s path integral p. 2

3 Classical action principle With L(ẋ, x, t) the system s Lagrangian, the classical action functional is defined by S [ x(t) ] = tb t a L(ẋ,x,t)dt. S is a mapping of the set of functions x(t) ( paths ) into the real numbers (hence the name functional ). Feynman s path integral p. 3

4 Classical action principle With L(ẋ, x, t) the system s Lagrangian, the classical action functional is defined by S [ x(t) ] = tb t a L(ẋ,x,t)dt. S is a mapping of the set of functions x(t) ( paths ) into the real numbers (hence the name functional ). The classical action principle now states that among all possible paths, the physical system actually chooses the particular path x a (t) that is associated with the least action δs [ x a (t) ] = 0. Feynman s path integral p. 3

5 Water v w v s Sand LS Life-saver problem as an example of a variation of a functional Feynman s path integral p. 4

6 From variational calculus, we find that the particular path x a (t) that minimizes the action S satisfies the Lagrange equation of motion d dt ( ) L ẋ L x = 0. Feynman s path integral p. 5

7 From variational calculus, we find that the particular path x a (t) that minimizes the action S satisfies the Lagrange equation of motion d dt ( ) L ẋ L x = 0. With the transition to quantum mechanics (QM), we encounter the following problems: The classical concepts of point particles and sharp trajectories do not exist anymore in QM QM is the more detailed theory we can infer from QM to classical mechanics but not vice versa. Feynman s path integral p. 5

8 From variational calculus, we find that the particular path x a (t) that minimizes the action S satisfies the Lagrange equation of motion d dt ( ) L ẋ L x = 0. With the transition to quantum mechanics (QM), we encounter the following problems: The classical concepts of point particles and sharp trajectories do not exist anymore in QM QM is the more detailed theory we can infer from QM to classical mechanics but not vice versa. There exist various quantization prescriptions aiming to deduce a quantum theory from an underlying classical theory. These prescriptions are cooking recipes. Feynman s path integral p. 5

9 Canonical quantization The classical Hamiltonian of a point particle in an external potential V (x) is given by H = p2 2m + V (x) = e, with e denoting the particle s energy. Feynman s path integral p. 6

10 Canonical quantization The classical Hamiltonian of a point particle in an external potential V (x) is given by H = p2 2m + V (x) = e, with e denoting the particle s energy. The canonical quantization then means to promote the canonical variables x,p,e into the state of operators ˆx, ˆp,ê that act on a wave function ψ(x,t) ) ) ψ (ˆxψ (x,t) = xψ(x,t), (ˆpψ (x,t) = i x, (êψ ) ψ (x,t) = i t. We thus derive the Schrödinger equation Ĥψ = ( 2 2m 2 x 2 + V (x) ) ψ = i t ψ. Feynman s path integral p. 6

11 The above procedure to derive the Schrödinger equation comprises of two steps (eqom=equation of motion) classical action integral classical eqom least action principle canonical quantization classical eqom quantum eqom Feynman s path integral p. 7

12 The above procedure to derive the Schrödinger equation comprises of two steps (eqom=equation of motion) classical action integral classical eqom least action principle canonical quantization classical eqom quantum eqom One may now ask: is there a way to directly derive the quantum equation for a given classical system from a modified formulation of the action principle? Feynman s path integral p. 7

13 The above procedure to derive the Schrödinger equation comprises of two steps (eqom=equation of motion) classical action integral classical eqom least action principle canonical quantization classical eqom quantum eqom One may now ask: is there a way to directly derive the quantum equation for a given classical system from a modified formulation of the action principle? The answer is: yes! Namely by means of Feynman s path integral quantization on the basis of the appropriate quantum formulation of the action principle: quantum action integral path integral quantum eqom Feynman s path integral p. 7

14 Quantum action principle According to the statistical interpretation of the wave function, ψ(x,t) 2 gives the probability density of finding the particle at the point x at time t. Its evolution can be described by the integral equation ψ(x b,t b ) = K(x b,t b ;x a,t a )ψ(x a,t a )dx a. Feynman s path integral p. 8

15 Quantum action principle According to the statistical interpretation of the wave function, ψ(x,t) 2 gives the probability density of finding the particle at the point x at time t. Its evolution can be described by the integral equation ψ(x b,t b ) = K(x b,t b ;x a,t a )ψ(x a,t a )dx a. Postulate: the transition amplitude K(b; a) is the sum of the contributions from all paths K(b;a) = all paths from a to b φ [ x(t) ], φ [ x(t) ] exp ( i S[ x(t) ] ). wherein S [ x(t) ] denotes the classical action functional. Feynman s path integral p. 8

16 K(b;a) = all paths from a to b φ [ x(t) ], φ [ x(t) ] exp ( i S[ x(t) ] ). Not only the path with the extreme action is relevant; rather it is that all paths contribute to the transition amplitude K(b; a). Feynman s path integral p. 9

17 K(b;a) = all paths from a to b φ [ x(t) ], φ [ x(t) ] exp ( i S[ x(t) ] ). Not only the path with the extreme action is relevant; rather it is that all paths contribute to the transition amplitude K(b; a). All paths contribute equal magnitude, i.e. all paths have the same weight. Feynman s path integral p. 9

18 K(b;a) = all paths from a to b φ [ x(t) ], φ [ x(t) ] exp ( i S[ x(t) ] ). Not only the path with the extreme action is relevant; rather it is that all paths contribute to the transition amplitude K(b; a). All paths contribute equal magnitude, i.e. all paths have the same weight. The paths differ in their phases. The phase of the contribution from a given path is the classical action S for that path in units of. Feynman s path integral p. 9

19 K(b;a) = all paths from a to b φ [ x(t) ], φ [ x(t) ] exp ( i S[ x(t) ] ). Not only the path with the extreme action is relevant; rather it is that all paths contribute to the transition amplitude K(b; a). All paths contribute equal magnitude, i.e. all paths have the same weight. The paths differ in their phases. The phase of the contribution from a given path is the classical action S for that path in units of. We will see that the classical action principle emerges as the classical limit of the quantum action principle. Feynman s path integral p. 9

20 Classical limit As is tiny, the phase S/ of a contribution is some very, very large angle. Feynman s path integral p. 10

21 Classical limit As is tiny, the phase S/ of a contribution is some very, very large angle. Small changes in an arbitrary path will, generally, make enormous changes in phase exp(is/ ) will oscillate rapidly between ±1. Feynman s path integral p. 10

22 Classical limit As is tiny, the phase S/ of a contribution is some very, very large angle. Small changes in an arbitrary path will, generally, make enormous changes in phase exp(is/ ) will oscillate rapidly between ±1. If the neighboring path has a different action, then their contributions will add to zero. Feynman s path integral p. 10

23 Classical limit As is tiny, the phase S/ of a contribution is some very, very large angle. Small changes in an arbitrary path will, generally, make enormous changes in phase exp(is/ ) will oscillate rapidly between ±1. If the neighboring path has a different action, then their contributions will add to zero. For the path in the vicinity of the classical path, we get no change in S (in the first order), since δs = 0. we get important contributions to K(b;a) only for paths in the vicinity of the classical path. Feynman s path integral p. 10

24 Classical limit As is tiny, the phase S/ of a contribution is some very, very large angle. Small changes in an arbitrary path will, generally, make enormous changes in phase exp(is/ ) will oscillate rapidly between ±1. If the neighboring path has a different action, then their contributions will add to zero. For the path in the vicinity of the classical path, we get no change in S (in the first order), since δs = 0. we get important contributions to K(b;a) only for paths in the vicinity of the classical path. In the limit 0, only the classical path contributes. Feynman s path integral p. 10

25 Riemann integral f(x) h x 0 x x x x x 1 2 i i+1 N x A i f(x i ), A = lim h 0 N h N i=0 f(x i ), h = x N x 0 N The step width h directly acts as the integration measure. Feynman s path integral p. 11

26 Path integral t t b t i+1 t i t a ε Nǫ = t b t a ǫ = t i+1 t i t a t 0 t b t N x a x 0 x b x N x a x x x i i+1 b We divide the independent variable time into steps of width ǫ we obtain a set of values t i between t a and t b We then select at each time t i some special point x i we thus construct a path by connecting all the points x i so selected with straight lines. Feynman s path integral p. 12 x

27 We finally sum the action exponential φ[x(t)] over all paths constructed in this manner K(b;a)... φ [ x(t) ] dx 1 dx 2...dx N 1. This definition corresponds to the Riemann integral definition A i f(x i ). Feynman s path integral p. 13

28 We finally sum the action exponential φ[x(t)] over all paths constructed in this manner K(b;a)... φ [ x(t) ] dx 1 dx 2...dx N 1. This definition corresponds to the Riemann integral definition A i f(x i ). The approximation to K(b;a) gets better by making ǫ smaller. Similar to the case of the Riemann integral, we need a normalization factor A(ǫ) in order for the procedure to converge K(b;a) = lim ǫ 0 N 1 A(ǫ) N... exp We encounter an infinite chain of integrals! ( i S[ x(t) ] ) dx 1 dx 2...dx N 1. Feynman s path integral p. 13

29 In order to distinguish a path integral from a conventional Riemann integral, a specific notation is appropriate K(b;a) = exp ( i S[ x(t) ] ) D(t), S [ x(t) ] tb = L(ẋ,x,t)dt. t a Feynman s path integral p. 14

30 In order to distinguish a path integral from a conventional Riemann integral, a specific notation is appropriate K(b;a) = exp ( i S[ x(t) ] ) D(t), S [ x(t) ] tb = L(ẋ,x,t)dt. t a From the definition of the path integral, we immediately conclude for the transition amplitude K(i + 1;i) of an infinitesimal time step ǫ K(i + 1;i) = 1 A(ǫ) exp ( i S[ i + 1,i ] ) with S [ i + 1,i ] = ǫl ( xi+1 x i ǫ, x i+1 + x i 2, t i+1 + t i 2 ). We will now work out some simple examples. Feynman s path integral p. 14

31 Example 1: Free particle The non-relativistic Lagrangian L of a single free particle is L(ẋ) = m 2 ẋ2. The quantum mechanical transition amplitude K(x b,t b ;x a,t a ) for a free particle over a finite time interval t b t a is then obtained by the path integral K(b;a)= lim ǫ 0 N 1 A(ǫ) N... exp ( im 2 N 1 i=0 ) (x i+1 x i ) 2 dx 1...dx N 1. ǫ Feynman s path integral p. 15

32 Example 1: Free particle The non-relativistic Lagrangian L of a single free particle is L(ẋ) = m 2 ẋ2. The quantum mechanical transition amplitude K(x b,t b ;x a,t a ) for a free particle over a finite time interval t b t a is then obtained by the path integral K(b;a)= lim ǫ 0 N 1 A(ǫ) N... exp ( im 2 N 1 i=0 ) (x i+1 x i ) 2 dx 1...dx N 1. ǫ The integrand is an exponential of a quadratic form ( gaussian ) The integral of a gaussian is again a gaussian. The chain of N 1 integrals can be solved analytically, and, consequently, the limit N. Feynman s path integral p. 15

33 The first two terms of the sum are integrated over x 1 ( im [ exp (x2 x 1 ) 2 + (x 1 x 0 ) 2]) dx 1 2 ǫ ( ) 2iπ ǫ 1 im = exp m 2 2 2ǫ (x 2 x 0 ) 2 Feynman s path integral p. 16

34 The first two terms of the sum are integrated over x 1 ( im [ exp (x2 x 1 ) 2 + (x 1 x 0 ) 2]) dx 1 2 ǫ ( ) 2iπ ǫ 1 im = exp m 2 2 2ǫ (x 2 x 0 ) 2 We add the third term to this result and integrate over x 2 = ( π ǫ im im exp [ ǫ (x 2 x 0 ) 2 + (x 3 x 2 ) 2]) dx 2 ( ) 2 ( ) 2iπ ǫ 1 im m 3 exp 2 3ǫ (x 3 x 0 ) 2 Feynman s path integral p. 16

35 The first two terms of the sum are integrated over x 1 ( im [ exp (x2 x 1 ) 2 + (x 1 x 0 ) 2]) dx 1 2 ǫ ( ) 2iπ ǫ 1 im = exp m 2 2 2ǫ (x 2 x 0 ) 2 We add the third term to this result and integrate over x 2 = ( π ǫ im im exp [ ǫ (x 2 x 0 ) 2 + (x 3 x 2 ) 2]) dx 2 ( ) 2 ( ) 2iπ ǫ 1 im m 3 exp 2 3ǫ (x 3 x 0 ) 2 After N 1 integrations, we have ( ) N 1 ( ) 1 2iπ ǫ 1 im K(b;a)= lim ǫ 0 exp N A(ǫ) N m N 2 Nǫ (x N x 0 ) 2 Feynman s path integral p. 16

36 For the limit ǫ 0,N to exist, we must define A(ǫ) as A(ǫ) = 2iπ ǫ m. In contrast to the Riemann integral, the measure of a path integral cannot be defined a priori, but must be chosen appropriately in order for the integral to converge. Feynman s path integral p. 17

37 For the limit ǫ 0,N to exist, we must define A(ǫ) as A(ǫ) = 2iπ ǫ m. In contrast to the Riemann integral, the measure of a path integral cannot be defined a priori, but must be chosen appropriately in order for the integral to converge. With Nǫ = t b t a,x N x b,x 0 x a, we thus find the final result ( ) m K(b;a) = 2iπ (t b t a ) exp im (x b x a ) 2. 2 t b t a Feynman s path integral p. 17

38 For the limit ǫ 0,N to exist, we must define A(ǫ) as A(ǫ) = 2iπ ǫ m. In contrast to the Riemann integral, the measure of a path integral cannot be defined a priori, but must be chosen appropriately in order for the integral to converge. With Nǫ = t b t a,x N x b,x 0 x a, we thus find the final result ( ) m K(b;a) = 2iπ (t b t a ) exp im (x b x a ) 2. 2 t b t a It is straightforward to verify that the transition amplitude K(b; a) satisfies the partial differential equation i K(b;a) t b = 2 2m 2 K(b;a) x 2 b K(b; a) satisfies the Schrödinger equation.. Feynman s path integral p. 17

39 Example 2: Schrödinger equation On the basis of the path integral formalism, it is actually possible to derive the Schrödinger equation. The starting point is the non-relativistic Lagrangian of a point particle of mass m in an external, possibly time-dependent potential V, L(ẋ,x,t) = m 2 ẋ2 V (x,t). Feynman s path integral p. 18

40 Example 2: Schrödinger equation On the basis of the path integral formalism, it is actually possible to derive the Schrödinger equation. The starting point is the non-relativistic Lagrangian of a point particle of mass m in an external, possibly time-dependent potential V, L(ẋ,x,t) = m 2 ẋ2 V (x,t). The modification of the particle s wave function ψ(x, t) along an infinitesimal time step ǫ is then described by the integral equation ψ(x,t + ǫ) = 1 { exp iǫ [ m 2 A(ǫ) ( ) 2 x y V ǫ ( x + y,t + ǫ 2 2) ]} ψ(y,t)dy. Feynman s path integral p. 18

41 After performing a change of the integration variable y = η + x, the integral simplifies to ψ(x,t + ǫ) = 1 A(ǫ) [ iǫ m 2 { exp dy = dη η 2 ǫ 2 V (x + η 2,t + ǫ 2 ) ]} ψ(x + η,t)dη. Feynman s path integral p. 19

42 After performing a change of the integration variable y = η + x, the integral simplifies to ψ(x,t + ǫ) = 1 A(ǫ) [ iǫ m 2 { exp dy = dη η 2 ǫ 2 V (x + η 2,t + ǫ 2 ) ]} ψ(x + η,t)dη. We expand the expressions for the wave function and for the potential ψ(x + η,t) = ψ(x,t) + η ψ(x,t) + η2 2 ψ(x,t) +... x 2 x 2 ψ(x,t + ǫ) = ψ(x,t) + ǫ ψ(x,t) +... V ( x + η 2,t + ǫ ) 2 = V (x,t) + η 2 t V (x,t) x + ǫ 2 V (x,t) t +... Feynman s path integral p. 19

43 and insert these expansions into the integral equation. Omitting all terms of higher order than one in ǫ, this yields ψ(x,t) + ǫ ψ t = 1 {(1 iǫ ) ( ) im A(ǫ) V ψ(x, t) exp 2 ǫ η2 dη [( + 1 iǫ ) 2 V ψ x iǫ ] V ψ ( ) } η 2 im 2 x x 2 exp 2 ǫ η2 dη. We have already skipped all integrals that are of odd power in the integration variable, η, since these integrals vanish. Feynman s path integral p. 20

44 and insert these expansions into the integral equation. Omitting all terms of higher order than one in ǫ, this yields ψ(x,t) + ǫ ψ t = 1 {(1 iǫ ) ( ) im A(ǫ) V ψ(x, t) exp 2 ǫ η2 dη [( + 1 iǫ ) 2 V ψ x iǫ ] V ψ ( ) } η 2 im 2 x x 2 exp 2 ǫ η2 dη. We have already skipped all integrals that are of odd power in the integration variable, η, since these integrals vanish. The gaussian integrals can now be solved analytically ψ(x,t) + ǫ ψ t = 1 A(ǫ) + [( 1 iǫ V ) 2 ψ x 2 iǫ { ( 1 iǫ V V x ) 2iπ ǫ ψ(x, t) m ( ) 3 } 2iπ ǫ. m ] ψ 1 x 4π Clearly, the equation must establish an identity for ǫ 0. Feynman s path integral p. 20

45 We thus obtain the condition to determine A(ǫ) ψ(x,t) = 1 2iπ ǫ A(ǫ) ψ(x,t) m A(ǫ) = 2iπ ǫ m. This is the measure for the infinitesimal path integral. Feynman s path integral p. 21

46 We thus obtain the condition to determine A(ǫ) ψ(x,t) = 1 2iπ ǫ A(ǫ) ψ(x,t) m A(ǫ) = 2iπ ǫ m. This is the measure for the infinitesimal path integral. The equation now simplifies to ψ+ǫ ψ (1 t = iǫ ) V ψ+ i ǫ [( 1 iǫ ) 2 2m V ψ x iǫ ] V ψ. 2 x x Feynman s path integral p. 21

47 We thus obtain the condition to determine A(ǫ) ψ(x,t) = 1 2iπ ǫ A(ǫ) ψ(x,t) m A(ǫ) = 2iπ ǫ m. This is the measure for the infinitesimal path integral. The equation now simplifies to ψ+ǫ ψ (1 t = iǫ ) V ψ+ i ǫ [( 1 iǫ ) 2 2m V ψ x iǫ ] V ψ. 2 x x The term ψ(x,t) drops out, and we can divide by ǫ ψ t = i V ψ + i [( 1 iǫ ) 2 2m V ψ x iǫ ] V ψ. 2 x x Feynman s path integral p. 21

48 We thus obtain the condition to determine A(ǫ) ψ(x,t) = 1 2iπ ǫ A(ǫ) ψ(x,t) m A(ǫ) = 2iπ ǫ m. This is the measure for the infinitesimal path integral. The equation now simplifies to ψ+ǫ ψ (1 t = iǫ ) V ψ+ i ǫ [( 1 iǫ ) 2 2m V ψ x iǫ ] V ψ. 2 x x The term ψ(x,t) drops out, and we can divide by ǫ ψ t = i V ψ + i [( 1 iǫ ) 2 2m V ψ x iǫ ] V ψ. 2 x x Taking to limit ǫ 0, we finally get ψ t = i V ψ + i 2m 2 ψ x 2, which obviously constitutes the Schrödinger equation. Feynman s path integral p. 21

49 Relativistic generalization The path integral based on non-relativistic Lagrangians L nr may be transposed to a relativistic description by 1. introducing the particle s proper time s as the new system evolution parameter, 2. treating the time t(s) as an additional dependent variable on equal footing with the configuration space variables q(s) commonly referred to as the principle of homogeneity in space-time 3. by replacing the conventional non-relativistic Lagrangian L nr with the corresponding Lorentz-invariant extended Lagrangian L 1, We will sketch this approach as the last topic of this talk. Feynman s path integral p. 22

50 The relativistic generalization of the integral equation for the space-time evolution of a wave function ψ(x,t) ψ(x µ ) is ψ(x b,t b ) = K(x b,t b ;x a,t a )ψ(x a,t a )d 4 x a, where x denotes the configuration space vector, and x µ the set of Minkowski space variables. Feynman s path integral p. 23

51 The relativistic generalization of the integral equation for the space-time evolution of a wave function ψ(x,t) ψ(x µ ) is ψ(x b,t b ) = K(x b,t b ;x a,t a )ψ(x a,t a )d 4 x a, where x denotes the configuration space vector, and x µ the set of Minkowski space variables. The analogous path integral description for the propagation of a wave function ψ(x µ ) along an infinitesimal space-time step is then ψ(x µ b ) = 1 A(ǫ) exp ( ) i S 1,ǫ ψ(x µ a)d 4 x a. Herein, S 1,ǫ denotes the extended action along the infinitesimal space-time step, which follows from the system s extended Lagrangian L 1 (dx µ /ds,x µ ) as S 1,ǫ = ǫl 1 ( x µ b xµ a ǫ, xµ b + xµ a 2 ),. Feynman s path integral p. 23

52 Example: relativistic point particle of mass m and charge ζ in an external electromagnetic field. The conventional relativistic Lagrangian of this system is L(ẋ,x,t) = mc 2 1 ẋ2 c + ζ A(x,t)ẋ ζφ(x,t). 2 c φ and A denote the scalar and the vector potentials, respectively. Feynman s path integral p. 24

53 Example: relativistic point particle of mass m and charge ζ in an external electromagnetic field. The conventional relativistic Lagrangian of this system is L(ẋ,x,t) = mc 2 1 ẋ2 c + ζ A(x,t)ẋ ζφ(x,t). 2 c φ and A denote the scalar and the vector potentials, respectively. Problem: the Lagrangian is no longer a quadratic form in the dynamic variables. The path integral formalism does not yield gaussian integrals. The path integral cannot be worked out. Feynman s path integral p. 24

54 The extended Lagrangian L 1 that desribes the same dynamics as the conventional relativistic Lagrangian L writes [ ( ) 2 ( ) 2 L 1 = 1 1 dq dt 2 mc2 1] + ζ c 2 ds ds c A(q,t)dq ζ φ(q,t)dt ds ds. The extended Lagrangian of a relativistic particle in an external electromagnetic field is quadratic in its velocities. Feynman s path integral p. 25

55 The extended Lagrangian L 1 that desribes the same dynamics as the conventional relativistic Lagrangian L writes [ ( ) 2 ( ) 2 L 1 = 1 1 dq dt 2 mc2 1] + ζ c 2 ds ds c A(q,t)dq ζ φ(q,t)dt ds ds. The extended Lagrangian of a relativistic particle in an external electromagnetic field is quadratic in its velocities. Based on this Lagrangian L 1, we may proceed similarly to the case of Schrödinger equation. Working this out, we end up with the Klein-Gordon equation ( x α iζ c A α )( x α iζ c Aα ) ψ = ( mc ) 2ψ. with A 0 φ and x 0 = ct,x 0 = ct. The Klein-Gordon equation is the quantum equation of a system whose classical counterpart is a relativistic point particle in an external electromagnetic field. Feynman s path integral p. 25

56 Conclusions It is possible to formulate a generalized action principle such that the fundamental equations of quantum physics can be derived from it. The classical action principle emerges from it for the limit 0. Feynman s path integral p. 26

57 Conclusions It is possible to formulate a generalized action principle such that the fundamental equations of quantum physics can be derived from it. The classical action principle emerges from it for the limit 0. This way, the Schrödinger equation emerges as the quantum equation of a system whose classical counterpart is given by a point particle in an external potential. Feynman s path integral p. 26

58 Conclusions It is possible to formulate a generalized action principle such that the fundamental equations of quantum physics can be derived from it. The classical action principle emerges from it for the limit 0. This way, the Schrödinger equation emerges as the quantum equation of a system whose classical counterpart is given by a point particle in an external potential. The path integral formalism thus replaces the rather unsatisfactory ad hoc prescriptions ( equivalence principle, canonical quantization ). Feynman s path integral p. 26

59 Treating space and time variables on equal footing, we may straightforwardly generalize the path integral formalism to cover relativistic quantum physics. Feynman s path integral p. 27

60 Treating space and time variables on equal footing, we may straightforwardly generalize the path integral formalism to cover relativistic quantum physics. On the basis of the extended Lagrangian of a relativistic point particle in an external electromagnetic field, we could show that the corresponding quantum equation is given by the Klein-Gordon equation. Feynman s path integral p. 27

61 Treating space and time variables on equal footing, we may straightforwardly generalize the path integral formalism to cover relativistic quantum physics. On the basis of the extended Lagrangian of a relativistic point particle in an external electromagnetic field, we could show that the corresponding quantum equation is given by the Klein-Gordon equation. This talk can be downloaded from struck Feynman s path integral p. 27

62 Treating space and time variables on equal footing, we may straightforwardly generalize the path integral formalism to cover relativistic quantum physics. On the basis of the extended Lagrangian of a relativistic point particle in an external electromagnetic field, we could show that the corresponding quantum equation is given by the Klein-Gordon equation. This talk can be downloaded from struck More details can be found in: Extended Hamilton-Lagrange formalism and its application to Feynman s path integral for relativistic quantum physics, Int. J. Mod. Phys. E. 18, No. 1 (2009) p Feynman s path integral p. 27