Return to Answer

deleted 1 characters in body

Source Link

edited Feb 26, 2010 at 3:31

Hi Theo,

Your definition is roughly correct, yes. For Wiener measure on paths in vector spaces, see Chapter 3 + Appendix A of the 2nd edition of Glimm & Jaffe. On curved targets, I think Bruce Driver has some good lecture notes. Two warnings One warning: the rough definition of Wiener measure is misleading in one way: Wiener measure is naturally constructed as a measure on a space of distributions which contains the continuous functions.
I don't think there's a general theory. Trying a Lagrangian of the form $F = (\dot{\phi})^{1000}$ will not result in happiness. But: You can define Wiener measure using the standard kinetic term $\int \frac{1}{2}|\dot{\phi}|^2dt$ for any $\hbar$, and you can safely add a potential $V$ which is bounded below and integrable to the kinetic term.
Any observable you can write down should give you a Wiener integrable function, in quantum mechanics. This is not true in QFT, however. Most of the work in constructive QFT is proving that the measure on the space of histories actually has moments!

Hi Theo,

Your definition is roughly correct, yes. For Wiener measure on paths in vector spaces, see Chapter 3 + Appendix A of the 2nd edition of Glimm & Jaffe. On curved targets, I think Bruce Driver has some good lecture notes. Two warnings: the rough definition of Wiener measure is misleading in one way: Wiener measure is naturally constructed as a measure on a space of distributions which contains the continuous functions.
I don't think there's a general theory. Trying a Lagrangian of the form $F = (\dot{\phi})^{1000}$ will not result in happiness. But: You can define Wiener measure using the standard kinetic term $\int \frac{1}{2}|\dot{\phi}|^2dt$ for any $\hbar$, and you can safely add a potential $V$ which is bounded below and integrable to the kinetic term.
Any observable you can write down should give you a Wiener integrable function, in quantum mechanics. This is not true in QFT, however. Most of the work in constructive QFT is proving that the measure on the space of histories actually has moments!

Hi Theo,

Your definition is roughly correct, yes. For Wiener measure on paths in vector spaces, see Chapter 3 + Appendix A of the 2nd edition of Glimm & Jaffe. On curved targets, I think Bruce Driver has some good lecture notes. One warning: the rough definition of Wiener measure is misleading in one way: Wiener measure is naturally constructed as a measure on a space of distributions which contains the continuous functions.
I don't think there's a general theory. Trying a Lagrangian of the form $F = (\dot{\phi})^{1000}$ will not result in happiness. But: You can define Wiener measure using the standard kinetic term $\int \frac{1}{2}|\dot{\phi}|^2dt$ for any $\hbar$, and you can safely add a potential $V$ which is bounded below and integrable to the kinetic term.
Any observable you can write down should give you a Wiener integrable function, in quantum mechanics. This is not true in QFT, however. Most of the work in constructive QFT is proving that the measure on the space of histories actually has moments!

Source Link

created Feb 25, 2010 at 23:05

Hi Theo,

Your definition is roughly correct, yes. For Wiener measure on paths in vector spaces, see Chapter 3 + Appendix A of the 2nd edition of Glimm & Jaffe. On curved targets, I think Bruce Driver has some good lecture notes. Two warnings: the rough definition of Wiener measure is misleading in one way: Wiener measure is naturally constructed as a measure on a space of distributions which contains the continuous functions.
I don't think there's a general theory. Trying a Lagrangian of the form $F = (\dot{\phi})^{1000}$ will not result in happiness. But: You can define Wiener measure using the standard kinetic term $\int \frac{1}{2}|\dot{\phi}|^2dt$ for any $\hbar$, and you can safely add a potential $V$ which is bounded below and integrable to the kinetic term.
Any observable you can write down should give you a Wiener integrable function, in quantum mechanics. This is not true in QFT, however. Most of the work in constructive QFT is proving that the measure on the space of histories actually has moments!