If you have listened to some of my online MP3 recordings you may have wondered why I prefer fade outs for most of the songs rather than regular endings.   There are two reasons for that.
Firstly there are for each style (bossa, ballad, jazz, etc.) two preprogrammed endings on my electronic keyboard. They are quite good but of course may not always be in character with the way I play a song. In fact in most cases they are not.
The second reason has a much deeper emotional and philosophical origin. I only realised this a few days ago when, most surprisingly, I was reading something in Brian Greene's 'The Fabric of the Cosmos' (Chapter 7, page 180). It makes a most interesting story I find. Here goes.

The Nobel laureate and brilliant 20th Century physicist Richard Feynman formulated a way of thinking about the uncertainty in quantum mechanics surrounding the properties and movements of the tiniest building blocks of our universe, which appear to be both particles and waves at the same time.

In his sum over histories principle Feynman states that "if there are alternative ways in which a given outcome can be achieved, then there is a sense in which the alternate histories all happen, and happen simultaneously."

Let me explain what this means. Imagine I am standing at the very Southern end of Mooloolaba beach, at the "Spit", the stone pier extending into the sea (just to the left of the four high rise buildings on the photo below).
I have had my swim, I am hungry and want to get back to the Surf Club (just to the right of the Life Savers watch tower in the right foreground of the photo) for lunch. There are four practical ways for me to get there :

jump into the surf and swim back walk back along the beach take the walking trail through the dunes behind the beach or walk back on the main road farther inland

But what if a tiny electron or a photon (light particle-wave) were to do the same thing ? Starting from the same point at the Spit where I walked back from it travels to the Surf Club in an instant (at the speed of light). But when it arrives it appears as if it has got there by using all four alternative paths : the surf, the beach, walking trail and main road (plus many other possible pathways) all at the same time *.   Weird ? Tell me about it ! But reality, as we are starting to discover now, is quite amazing, and very different from what we have always believed it to be.

Back to music. Strangely enough I see a clear parallel between Feynman's sum over histories and the creativity of Jazz improvisation.   A song generally consists of two components, a melody line and an underlying succession of chords (a socalled "chord progression") which give the song colour and mood. The melody and chords of course do not clash but are in harmony with one another.
Some composers start by writing the melody first, then construct an appropriate chord progression for it. Others (like me mostly) start with formulating a chord progression and then create a melody line over it. Whichever way it is created once a song exists it is its chord progression which represents the fundamental essence of the song.

Over a song's chord progression we can create an infinite number of other melodies, the only restraint being that a melody should not clash but be in harmony with the chord progression. This is exactly what the Jazz improvisor does. He creates an instant new melody over the given chord sequence, within the constraints of his musical knowledge and understanding, technical skills and creative imagination. Every performer will have his own unique approach to a song and over time, as his familiarity and emotional insight into the song increases, his improvisations will improve in terms of meaning and subtlety.
One of my teachers, the brilliant and absolute world class Australian saxophonist Barry Duggan told me once that it had taken him 10 years (!!) to come fully to grips with Duke Ellington's composition Satin Doll. Revealing a glimpse of Barry's depth of insight and emotional involvement in his music.

So like in Feynman's sum over histories principle, there are numerous improvised melodic pathways from a song's beginning to its end. But since we are no tiny electrons or photons but large clumsy human beings we cannot perform all possible pathways at the same time. Instead we must be content to play our improvised melodies one after the other.
But as there are an infinite number of possible improvisations over a song, a song never ever really ends. That is why I much prefer to fade out at the end creating for you, the listener, the illusion that the song continues to go on, and that only I, the performer, gradually move away into the distance and out of the range of your ears.

The nearest thing (in terms of musical performance) to the quantum sum over histories theory is probably the group improvisation by Trad Jazz bands (we called them "Dixieland bands" in Holland), where trumpet, trombone, clarinet, bas, drums, piano, banjo each follow there own unique path through the chord progression of a song, while all playing simultaneously.